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'""o University LIBRARIES Stanford university LIBP""'ES
MATHEMATICAL PAPEKS.
lonlion: C. J. CLAY & SONS,
CAMBBIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
Camfmlige: DEIGHTON, BELL AND CO.
!Lrqj}ts : F. A. BROCEHAUS.
THE COLLECTED
MATHEMATICAL PAPERS
OF
AETHUE CAYLEY, Sc.D., F.E.S.,
8ADLERIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE.
VOL. II.
CAMBRIDGE :
AT THE UNIVERSITY PRESS.
1889
[All Rights reserved.^
9y^i_\o^^
CAMBRIDGE :
PRINTED BY C. J. CLAT, ILA. AMD SONS,
AT THE UNIVEBSITY PRESS.
ADVEETISEMENT.
THE present volume contains fiftyeight papers (numbered 101, 102,..., 158)
originally published, all but two of them, in the years 1851 to 1860:
they are here reproduced nearly but not exactly in chronological order.
The two excepted papers are 142, Numerical Tables Supplementary to
Second Memoir on Quantics, now first published (1889); and, 143, Tables
of the Covariants M to W of the Binary Quintic : from the Second,
Third, Fourth, Fifth, Eighth, Ninth and Tenth Memoirs on Quantics
(arranged in the present form, 1889) : the determination of the finite
number, 23, of the covariants of the quintic was made by Gordan in the
year 1869, and the calculation of them having been completed in my
Ninth and Tenth Memoirs, it appeared to me convenient in the present
republication to unite together the values of all the covariants : viz. those
of A to L are given in the Second Memoir 141, and the remainder
M to W in the paper 143. I have added to the Third Memoir 144, in the
notation thereof, some formulae which on account of a difference of notation
were omitted fi:om a former paper, 35.
I remark that the present volume comprises the first six of the ten
Memoirs on Quantics, viz. these are 139, 141, 144, 155, 156 and 158.
I have, in the Notes and References, inserted a discussion of some length
in reference to the paper 121, Note on a Question in the Theory of
Probabilities: and also some remarks in reference to the theory of Dis
tance developed in the Sixth Memoir on Quantics, 158.
C. II.
CONTENTS.
PAOB
101. Notes on Lagrange's Theorem ....... 1
Gamb. and DubL Math. Jour. t. vi. (1851), pp. 37 — 45
102. On a Double Infinite Series ....... 8
Camb. and DubL Math. Jour, t vi. (1851), pp. 45 — 47
103. On Certain Definite Integrals 11
Camb. and DubL Math. Jour, t vl (1851), pp. 136—140
104. On the Theory of Permutants 16
Camb. and DubL Math. Jour, t vii. (1852), pp. 40 — 51
105. Correction to the Postscript to the Pa/per on Permutants 27
Camb. and DubL Math. Jour, t vii. (1852), pp. 97—98
106. On the Singularities of Surfaces 28
Camb. and DubL Math. Jour. t. vii. (1852), pp. 166—171
107. On the Theory of Skew Surfaces 33
Camb. and DubL Math. Jour, t vii. (1852), pp. 171—173
108. On certain Multiple Integrals connected with the Theory of
Attractions 35
Camb. and DubL Math. Jour. t. vii. (1852), pp. 174—178
109. On the Rationalisation of certain Algebraical Equ/Uions . 40
Camb. and DubL Math. Jour. t. viii. (1853), pp. 97—101
110. Note on the Transformation of a TrigonomstHcal Expression . 45
Camb. and DubL Math. Jour. t. ix. (1854), pp. 61 — 62
111. On a Theorem of M. LejeuneDirichlet' s 47
Camb. and DubL Math. Jour. t. ix. (1854), pp. 163—165
&2
VUl CONTENTS.
PAOB
112. Demonstration of a Theorem relating to the Products of Sums
of Squares 49
Phil. Mag. t IV. (1852), pp. 516—519
113. On the Geometrical Representation of the Integral
Idx^Jix + a) {x + h) {x + c) 53
Phil. Mag. t. V. (1853), pp. 281—284
114. Analytical Researches connected with Steiner's Extension of
MalfattHs Problem 57
Phil. Trans, t cxlii. (for 1852), pp. 253—278
115. Note on the Porism of the Inandcircum^crihed Polygon . 87
Phil. Mag. t. VI. (1853), pp. 99—102
116. Correction of two Theorems rekuing to the Inandcircum^
scribed Polygon ......... 91
Phil. Mag. t VI. (1853), pp. 376—377
117. Note on the Integral \dx'^J{m'x){x + a){x + h){x + c) . . 93
Phil. Mag. t VL (1853), pp. 103—105
118. On the Harmonic Relation of two Lines or two Points . . 96
Phil. Mag. t. VI. (1853), pp. 105—107
119. On a Theorem for the Development of a Factorial . . . 98
Phil. Mag. t. VI. (1853), pp. 182—185
120. Note on a Generalisation of the Binomial Theorem . . 101
Phil. Mag. t VI. (1853), p. 185
121. Note on a Question in the Theory of Prohahilities . . . IDS
Phil. Mag. t VI. (1853), p. 259
122. On the nomographic Transformation of a Surface of tJie Second
Order into Itself ....... . . 10 ^^^
Phil. Mag. t'vi. (1853), pp. 326—333
123. On the Geometrical Representation of an Abelian Integral . 11 —
PhiL Mag. t vi. (1853), pp. 414—418
124. On a Property of the Caustic by Refraction of the Circle . 1
Phil. Mag. t VL (1853), pp. 427—431
CONTENTS.
IX
125. On the Theory of Groups as depending on the Symbolical
. Equation ^=1
Phil. Mag. t. VII. (1854), pp. 40—47
126. On the theory of Groups as depending on the Symbolical
Equation ^ = 1. Second Part
Phil. Mag. t VII. (1854), pp 408—409
127. On the nomographic Transformation of a Surface of the
Second Order into itself
PhU. Mag. t VII. (1854), pp. 208—212: continuation of 122
128. Developments on the Porism of the Inanddrcumsciibed Polygon
PhiL Mag. t. vii. (1854), pp. 339—345
129. On the Porism of the InandcircuToscribed Triangle^ and on
an irrational Transformation of two Ternary Quadratic
Forms each into itself. .......
Phil. Mag. t. IX. (1855), pp. 513—517
130. Deuxihne MSmoire sur les Fonctions doublement PSriodiques .
Liouville, t. xix. (1854), pp. 193—208 : sequel to 25
131. Nouvelles Recherches sur les Covariants
CreUe, t xlvii. (1854), pp. 109—125
132. RSponse d, une Question proposSe par M. Steiner
Crelle, t l. (1855), pp. 277—278
133. Sur un Thiorhme de M. Schlafli ......
Crelle, t. l. (1855), pp. 278—282
134. Remarques sur la Notation des Fonctions Algebriques
Crelle, t. l. (1855), pp. 282—285
135. Note sur les Covariants d'une Fanction Quadratique^ OubiquCy
ou Biquadratique d, deux IndSterminSes ....
Crelle, t l. (1855), pp. 285—287
136. Sur la Transformation d'une Fonction Quadratique en elle
mime par des Substitutions Ihieaires .....
Crelle, t l. (1855), pp. 288—289
137. Seeker ches Ult&i^ieures sur les Determinants gauches
Crelle, t. l. (1855), pp. 299—313: continuation of 52 and 69.
PAOB
123
131
133
138
145
150
164
179
181
185
189
192
202
CONTENTS.
PAOB
138. Recherches sur les Matrices dont hs termes sont des fonctions
linSaires d^une seule Ind4tei^min4e . . . . . 216
Crelle, t. l. (1855), pp. 313—317
139. An Introductory Memoir on Quantics 221
Phil. Trans, t cxLiv. (for 1864), pp. 244—258
140. Researches on the Partition of Numbers 235
Phil. Trans, t. cxlv. (for 1855), pp. 127—140
141. A Second Memoir on Quantics 250
Phil. Trans, t. cxlvi. (for 1856), pp. 101—126
142. Numerical Tables Supplementary to Second Memoir on Quantics 276
Now first published (1889)
143. Tables of the Covariants M to W of the Binary Quintic : from
the Second, Third, Fifth, Eighth, Ninth and Tenth Memoirs
on Quantics 282
Arranged in the present form, 1889
144. A Third Memmr on Quantics 310
PhiL Trans, t. cxlvi. (for 1856), pp. 627—647
145. A Memoir on Caustics 336
Phil. Trans, t. cxlvii. (for 1857), pp. 273—312
146. A Memoir on Curves of the Third Order .... 381
Phil. Trans, t cxlvii. (for 1857), pp. 415—446
147. A Memoir on the Symmetric Functions of the Roots of an
Equation 417
Phil. Trans, t. cxlvii. (for 1857), pp. 489 — 496
148. A Memxdr on the Resultant of a System of two Equations . 440
Phil. Trans, t. cxlvii. (for 1857), pp. 703—715
149. On the Symmetric Functions of the Roots of certain Systems
of two Equations ......... 454
Phil. Trana t. cxlvii. (for 1857), pp. 717—726
150. A Memoir on the Conditions for the Existence of given Systems
of Equalities among the Roots of an Equation . . . 465
Phil. Trans, t. cxlvil (for 1857), pp. 727—731
CONTENTS. XI
PAOX
51. Tables of the Sturmian Functions for Equations of the Second^
Thirds Fourth^ and Fifth Degrees 471
Phil. Trans, t cxLVii. (for 1857), pp. 733—736
52. A Memoir on the Theory of MatHces 475
Phil. Trans, t. cxLViii. (for 1858), pp. 17—37
)3. A Memoir on the Automorphic Linear Transfoi^mation of a
Bipartite Quadric Function 497
Phil. Trans, t. cxlviii. (for 1858), pp. 39—46
)4. Supplementary Researches on tJie Partition of Numbers . . 506
PhiL Trans, t. cxlviii. (for 1858), pp. 47—52
)5. A Fourth Memoir on Qualities 513
Phil. Trans, t. cxlviii. (for 1858), pp. 415—427
)6. A Fifth Memoir on Qiiantics . . . 527
Phil. Trans, t. cxlviil (for 1858), pp. 429—460
)7. On the Tangential of a Cubic 558
PhiL Trans, t. cxlviil (for 1858), pp. 461—463
8. A Sixth Memoir on Quantics 561
Phil. Trans, t cxLix. (for 1859), pp. 61—90
and References .......... 593
CLASSIFICATION.
Geometry
Theory of Distance, 158
Surfaces, 106, 107
Transformation of Qnadric Surfaces, 122, 127, 129, 136, 153
Steiner's extension of Mal&tti's Problem, 114
Inandcircumscribed triangle and polygon, 115, 116, 128, 129
Harmonic relation of two lines or points, 118
Question proposed by Steiner, 132
Caustics, 124, 145
Cubic Curves, 146, 157
Analysis
Skew Determinants, 137
Attractions and Multiple Integrals, 108
Definite Integrals, 103
Elliptic and Abelian Integrals, 110, 113, 117, 123, 130
Covariants, Quantics <fec., 131, 134, 135, 139, 141, 142, 143, 144, 155, 156, 158
Matrices, 138, 152
Partition of Numbers, 140, 154
Symmetric Functions dec., 147, 148, 149, 150
Lagrange's Theorem, 101
Double Infinite Series, 102
Permutants, 104, 105
Rationalisation of Algebraic Expression, 109
Transformation of Trigonometrical Expression, 110
Theorem of LejeuneDirichlet's, 111
Products of Sums of Squares, 112
Factorials, 119
Generalisation of Binomial Theorem, 120
Question in Probabilities, 121
Groups, 125, 126
Theorem of Schlafli's, on Elimination, 133
Sturmian Functions, 151
101]
101.
NOTES ON LAGRANGE'S THEOREM.
[From the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 37 — 45.]
I.
If in the ordinary form of Lagrange's theorem we write {x + a) for x, it becomes
X = hf(a + x),
F(a + x)=^Fa+jF'afa + &c (1)
It follows that the equation
F{a + x) = Fa + \j~^^{F'a/a)i (2)
must reduce itself to an identity when the two sides are expanded in powers of x\
A
da
(3)
or writing for shortness F, f instead of Fa, fa, and S for r , we must have
(where p extends from to r). Or what comes to the same,
[r]^^°M p[P>?[rp]^[>ir' ^^^^"^^} ^*>
where s extends from to (^ — p). The terms on the two sides which involve Z^F
are immediately seen to be equal ; the coefficients of the remaining terms S'jP on tht^
second side must vanish, or we must have
o. II,
1
s.^
2 NOTES ON Lagrange's theorem. [101
(s being less than r). Or in a somewhat more convenient form, writing p, q and it
for p — 8, r—p and r — «,
where 8 is constant and p and q vary subject to p+q = ky k being a given constant
diflferent from zero (in the case where A:=0, the series reduces itself to the single
term  ). The direct proof of this theorem will be given presently.
II.
The following symbolical form of Lagrange's theorem was given by me in the
Mathematical Journal, vol. ill. [1843], pp. 283—286, [8].
If x^a^hfx, (7)
then
Suppose /r = ^ (6 + h^sc\ or a? = a + A^ (6 + ky^x), then
Fx= (jY'^'^F'a c*«<*+*^».
But
(In fact the two general terms
{<t>{b + kita)}'^ and (^)*^e*^(<^)"',
of which the former reduces itself to e db(if)by^, are equal on account of the equiva
lence of the symbols
e**"^ and (A)*^e*^).
Hence
X = a •\ h4> (b •\ kyfrx), (8)
^=(i)*''"(^)'"^'«^*"'"^
and the coefficient of h^k^ is
w
r^(^r^'«<^'^)"(s)"('^>"
101] NOTES ON Lagrange's theorem. 3
A similar formula evidently applies to the case of any finite number of functions
0, y^, &c. : in the case of an infinite number we have
or the coefficient of h^kH^ ... is
mrmrY ■ :. (r«)" '" ■ {mT <♦') ■ (s)' <+'>• ■ •
the last of the series m, n, p being always zero; e.g. in the coefiicient of h^k^,
account must be had of the factor fij ('^V* or (^c)**. The above form is readily
pn)ved independently by Taylor s theorem, without the assistance of Lagrange's. If in it
we write h = k, &c., a = 6 = &c., and ^ = '^ = &c. =/, we have F (a + hf{a + hf{a +...) = Fx,
where ar = a + hfx. Hence, comparing the coefficient of A* with that given by Lagrange's
theorem,
whtre m + w+&c. = «, and as before Fa, /(/, . have been replaced by F^ / 8. By
comparing the coefficients of h^F,
where n ■¥p\... =<, the last of the series n, p ,., always vanishing. The formula (10)
deduced, as above mentioned, from Taylors theorem, and the subsequent formula (11)
with an independent demonstration of it, not I believe materially different from that
which will presently be given, are to be found in a memoir by M. Collins (volume ii.
(1H38) of the Memoirs of the Academy of St Petersburg), who appears to have made
very extensive researches in the theory of developments as connected with the combina
torial analysis.
in.
To demonstrate the formula (6), consider, in the first place, the expression
s^^^,[m''^')m^^)\<
ii^here j9 + 5 = it. Since
1_ _1 / 1 1_ __\
1—2
4 NOTES ON Lagrange's theorem. [101
this is immediately transformed into
= J 5 ^j?j^j, {^ (p + 1) (p + » + !)(«>' .y>*'8/)(8«/'».)
iu which last expression p + 9 = (^ — 1 ). Of this, after separating the factor Sf, the
general term is
[/>?[?«]* *p <^ + * + ^ <^^> (^^z^^)
equivalent to
I f^ «' V « f^5f {* (p + a + 1) (p + * + a + 1) (SV^+'+) (S^/^— ^0
 *p (p + « + ^ (s^y*^*) W'"'''^')]*
in which last expression p + g=A;a — 1. By repeating the reduction j times, the
general term becomes
1 1
A;(ifcal)(ifca/82) ... [a]*j8]^...
s+»/.y +'/...
x[p + » + ^ + a + /S...+jip^ (SPfP+'^^fi") {B^fP • » i  ^ •)},
where the sums a + /3... contain / terms,/ being less than j or equal to it, and S
extends to all combinations of the quantities a, /3... taken / and / together (so that
the summation contains 2^ terms). Also p43 = A; — a — )8... (J terms) — j, and the
products A; (A;  a  1 ) (Jfc  o  )8  2) . . . and [a]* [fif, . . . S*+y . ^ +y. . . contain each of
them j terms. Suppose the reduction continued until A: — a — )8 . . . ( j terms) — j = 0, then
the only values of p, q are p = 0, ? = ; and the general term of
becomes
1 1 8"+»/" ifi+^f t'i*
kikal){ka^2)... [aj« [/SJ". . . •'•* •'•^
I] NOTES ON Lagrange's theorem.
If ^ =s 0, the general term reduces itself to
mee finally, if ^ = —  , the general term of
omes
1 it is readily shown that the sum contained in this formula vanishes, which proves
equation in question.
IV.
The demonstration of the equation (11) is much simpler. We have
t is,
"e n extends from n = 1 to n = ^. Similarly
&C.
ence, substituting successively, and putting t — n—p — q = r, &c.,
d the last of these corresponding to a zero value of the last of the quantities
. is evidently the required equation (11).
V.
formula (18) in my paper on Lagrange's theorem (before referred to) is incorrect.
at present, after giving the proper form of the formula in question, to
the result of the substitution indicated at the conclusion of the paper. It
onvenient to call to mind the general theorem, that when any number
6 NOTES ON LAGRANQJi's THEOREM. [lOl
of variables a?, y, z .,. are connected with as many other variables t/, r, ti; . . . by
the same number of equations (so that the variables of each set may be considered
as functions of those of the other set) the quotient of the expressions dxdy ... and
dudv ... is equal to the quotient of two determinants formed with the functions which
equated to zero express the relations between the two sets of variables ; the former
with the differential coefficients of these functions with respect to a, » . . . , the latter
with the differential coefficients with respect to x, y Consequently the notation
^T may be considered as representing the quotient of these determinants. This
being premised, if we write
X'u — h0(x, y . . .) = 0,
yV'k<f>(x, y ...) = 0,
then the formula in question is
if for shortness the letters 0, 0, . . . , F denote what the corresponding functions become
when M, v, ... are substituted for x, y, — Let r denote the value which , j '" ,
A dudv...
considered as a function of x, y . . . , assumes when these variables are changed into
w, V, . . . , we have
V =
1AM, AS^d... !.
— KOu^» 1 — A?Ov0 . . .
By changing the function F, we obtain
Fix, y...) = S«***S^*^*...«*^** .FV;
where, however, it must be remembered that the A, A:, ... , in so far as they enter into
the function V, are not aflfected by the symbols AS^, AS*,... In order that we may
consider them to be so affected, it is necessary in the function V to replace A, A:, &c.
h k
by ^ , ^ , &c. Also, afler this is done, observing that the symbols ASu^, hB„0 ... affect
Ou Off
a function gW+uh ... /^ xh^ symbols hBuO, hB„0,.,. may be replaced by S,/, S/, ..., where
the is not an index, but an affix denoting that the differentiation is only to be
performed with respect to u, t; ... so far as these variables respectively enter into
the function 0. Transforming the other lines of the determinant in the same manner,
and taking out from Su "K * ••• the factor SuB^ ... in order to multiply this last
factor into the determinant, we obtain
Fij, y...) = S„"» S."'' . . . ««*♦• F D ;
where
n= ««««*. ««♦.... ,
101] NOTES ON Lagrange's theorem. 7
in which expression S„, S„... are to be replaced by
The complete expansion is easily arrived at by induction, and the form is somewhat
singular. In the case of a single variable u we have □ = Su, in the case of two
variables, □ = S„'S„' + S„'S„^ + K^K^ Or writing down only the aflSxes, in the case of
a single variable we have F] in the case of two variables FF, Fd, <f)F] and in the
case of three variables FFF, <f>FF, x^^y ^X^> ^^^> ^^^. ^^<^» ^^^> ^H* ^X^» 0^0.
XF<^, 4>F0, xxF^ ^X^> X^^'i where it will be observed that 6 never occurs in the
first place, nor ^ in the second place, nor d, <f> (in any order) in the first and second
places, &c., nor 0, (f>, x 0^ *^y order) in the first, second, and third places. And the
same property holds in the general case for each letter and binary, ternary, &c.
combination, and for the entire system of letters, and the system of affixes contains
every possible combination of letters not excluded by the rule just given. Thus in the
case of two letters, forming the system of aflBxes FF, F0, tf>F, 0F, F<\>, 0<f>, <f)0, the last
four are excluded, the first three of them by containing in the first place or <f>
in the second place, the last by containing <l>, in the first and second places : and
there remains only the terms FF, F0, ^F forming the system given above. Substituting
the expanded value of □ in the expression for F (a;, y...), the equation may either be
permitted to remain in the form which it thus assumes, or we may, in order to
obtain the finally reduced form, after expanding the powers of A, A: . . . , connect the
symbols S^*, S„*...Su', &c. with the corresponding functions 0, <f).,.F, and then omit the
affixes ; thus, in particular, in the case of a single variable the general term of Fx is
(the ordinary form of Lagranges theorem). In the case of two letters the general
term of F{xy y) is
(see the MScaniqvs Celeste, [Ed. 1, 1798] t. i. p. 176). In the case of three variables,
the general term is
MWW' ^'^'^''"*"''"' {^*V«AS^ +...}.
the sixteen terms within the { } being found by comparing the product S„SpS«, with
the system FFF, 4>FF^ &c., given above, and then connecting each symbol of diff'eren
tiation with the function corresponding to the aflfix. Thus in the first term the
^> ^vt ^vi ^a^b affect the F, in the second term the h^ affects ^^, and the 8^ and h^
each affect the F, and so on for the remaining terms. The form is of course deducible
from Laplace's general theorem, and the actual development of it is given in Laplace's
Memoir in the Hist, de VAcad. 1777. I quote from a memoir by Jacobi which I take
this opportunity of referring to, "De resolutione equationum per series infinitas,"
CreUe, t. vi. [1830], pp. 257 — 286, founded on a preceding memoir, "Exercitatio algebraica
circa discerptionem singularem fractionum quae plures variabiles involvunt," t. V. [1830],
pp. 344—364.
Stone Buildings, April 6, 1850.
8
[102
102.
ON A DOUBLE INFINITE SERIES.
[Fn)m the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 45 — 47.]
The following completely paradoxical investigation of the properties of the fianction
r (which I have been in possession of for some years) may perhaps be found interesting
from its connexion with the theories of expansion and divergent serieis.
Let Sr^r denote the sum of the values of ifyr for all integer values of r from
— X to X . Then writing
w = 2^[nl]'a:«»^, (1)
(where n is any number whatever), we have immediately
^' = 2r [n  ly^^ af*^ ^^rin If x^"''' = u ;
1 . du ^
that IS, ^ ~ ^» ^' ^ ^ ^»»^»
(the constant of integration being of course in general a function of n). Hence
(7n^ = 2r[nl]'*ic~^^; (2)
or 6* is expanded in general in a doubly infinite necessarily divergent series of /fractional
powers of x, (which resolves itself however in the case of n a positive or negative
integer, into the ordinary singly infinite series, the value of Cn in this case being
immediately seen to be Fn).
The equation (2) in its general form is to be considered as a definition of the
function (?». We deduce from it
Xr [n  1]' (flur)'»>^ = C„e«* ,
1^ [n'  ly {aaff^^' = Cn'^ ;
102] ON A DOUBLE INFINITE SERIES. 9
and also
2jk [n + n' . . .  1]* {a (a? + a?' . . . )}«+n'...i* = (?«+«•... c«<«+^ •>.
Multiplying the first set of series, and comparing with this last,
Cn+n' ...2.. ^ ... [n  l]* [n' ^ly... a;~^ a/«»^ . . .
= CnCn'...[n + n'...l]*(a; + a;'. ..)•'"'**'•""'"*, (3)
(where r, r denote any positive or negative integer numbers satisfying r + r'+...=A:+l— p,
p being the number of terms in the series n, n\...). This equation constitutes a
multinomial theorem of a class analogous to that of the exponential theorem contained
in the equation (2).
In particular
C^n' ... 2,y ... [niy [n'  ly . . . = CnC^. . . . [n + n' . . .  l]*,»+«'...it, (4)
and if p = 2, writing also m, n for n, n', and k—l — r for r*,
C„+,2,[ml]'[«l]*''=0„C„[m + nl]*2»+"'* (5)
or putting k = and dividing,
C„(7„C„+„ = 2„^2,[ml]'[nl]''. (6)
Now the series on the second side of this equation is easily seen to be convergent
(at least for "positive values of m, n). To determine its value write
then
(m, n) = I af^^ (1 — w)^^ dx ;
J
F(m, n)= I a?^»(l~ir)«*(ir+ f af"^ {I  xf^^ dx \
J Q J Q
and by successive integrations by parts, the first of these integrals is reducible to
1
^_^^Y 2r [w — l]*" [^ — l]"^"', ^ extending from — 1 to — x inclusively, and the second to
^^li^^^zi ^r [^ — l]*" [w — l]""*~^i ^* extending from to oo ; hence
or C„C„.C^„ = F(m, n) (7)
C. 11. 2
10 ON A DOUBLE INFINITE 8ERI1*>$. [102
which proves the identity of Cn^ with the function T (m), {Substituting in two of the
preceding equations, we have
TnTn'. . .  T (n + r' . . .) = [„+„'... _l]fcyn^n...t ^ry... [n  1]' [n'  1]' (8)
(where, as before, p denotes the number of terms in the series n, n\... and r+r'4...=ifc+l— /)),
the first side of which equation is, it is well known, reducible to a multiple definite
integral by means of a theorem of M. Dirichlet's. And
''("'• ") = [,« + niy 2"^»— * ^' ^"'  ^^' t»  1]*— ^ (9) ,
where r extends from — x to 4 « , and k is arbitrary. By giving large negative
values to this quantity, very convergent series may be obtained for the calculation of
F(m, n)].
103]
11
103.
ON CERTAIN DEFINITE INTEGRALS.
[From the Cambridge and Dvblin Mathematical Journal, vol. vi. (1851), pp. 136 — 140.]
Suppose that for any positive or negative integral value of r, we have ^(rc + ra)
= Ur yp'Xy Ur being in general a function of x, and consider the definite integral
J — 00
'a being any other function of x. In case of either of the functions yp'x, 'Vx becoming
iTiiinite for any real value a of x, the principal value of the integral is to be taken,
that is, 7 is to be considered as the limit of
(j + f* '^irx^xdx, (€ = 0),
a»n.d similarly, when one of the functions becomes infinite for several of such values
We have
/ r(r+i)a \
/ = (... I +...j^a?^«da?;
^^ changing the variables in the different integrals so as to make the limits of each
^» O, we have
1=1 [2'^(ic + ra)^(a; + ra)]da?,
Jo
"^ Extending to all positive or negative integer values of r, that is,
I=ryftx[XUr'ir{x + ra)]dx, (A)
Jo
2—2
i
12
ON CERTAIN DEFINITE INTEGRALS.
[103
which is true, even when the quantity under the integral sign becomes infinite for
particular values of x, provided the integral be replaced by its principal value, that is,
provided it be considered as the limit of
or
where a, or one of the limiting values a, 0, is the value of x, for which the quantity
under the integral sign becomes infinite^ and 6 is ultimately evanescent.
In particular, taking for simplicity a = tt, suppose
^ (a? 4 tt) = ± ^x, or ^ (a? + nr) = {±y y^x ;
then observing the equation
Z —  — = cot X, or = cosec x,
x + nr
according as the upper or under sign is taken, and assuming 'Vx = x~*^, we have finally
the former equation corresponding to the case of ^ (j? + tt) = y^x^ the latter to that of
'^ (ic 4 tt) = — y^x.
Suppose y^^x = yp'gx, g being a positive integer. Then
r * yjr^xdx _ ^ f * y^xdx
also if >fr (j; + w) = i^jr, then ^^ (;c + tt) = >^,a: ; but if ^/r (.c + tt) = — i^J, then '^^{x^ir)
= ±ylr^x, the upper or under sign according as g is even or odd. Combining these
equations, we have
y^(x{7r) = yp'X, g even or odd,
•^ (a; 4 tt) = — '^a?, g even.
' cot x] At = < j,^  j' ^x [(J^J" ' cot x] .ir ;
/.:
fgxdx _(^y^ f
'^ (a? 4 "w) = — ^a?, g odd,
g^'^\x
T 1 cosec
/.^=S>V/>« 0«^^^^^ ^^(..^'IJ^x [(^"'cosecx] d,.
103] ON CERTAIN DEFINITE INTEGRALS. 13
In particular
sinxdx
f Sin a
J 00 *^
/ sin ^07 f T J cot j: cir = (/**"* / sina: (jj coseca? ch, g even,
I sin^ra: fjj coseeo: c2a?= 9^"^] '^^^^ (;/") coseca: cir, jr odd,
I sin gxcotxdx^Tr, g even,
I sin ^a: cosec iTcia? = TT, g odd,
C'tSLUxdx ^ «
= 0, i&c.,
the number of which might be indefinitely extended.
The same principle applies to multiple integrals of any order: thus for double
integrals, if '^(x + ra, y + rb) = Ur ,i'^ (a:, y), then
I i ir{x, y)^(x, y)dxdy^r ( fix, y) 2 £7,,, ^ (a? + ra, y + sb). ... (B)
J— 3oJao J J
In particular, writing WyV for a, 6, and assuming y^(x + rWf y + sv) = (±y {±y yfr (x, y);
also '^(x, y) = (x + iy)~'^, where as usual i=\/ — 1»
where
a/ . • N x' (±X(±)'l
^ ^^ (x + ty {rw + sm)
S extending to all positive or negative integer values of r and s. Employing the
notation of a paper in the Cambridge Mathematical Journal, "On the Inverse Elliptic
Functions," t. iv. [1845], pp. 257 — 277, [24], we have for the diflferent combinations
of the ambiguous sign,
^/ • V iS(x+iy) 1
1. , , e(a? + ty)= , 7'\ =j./r'\>
^ ^^ y{x + %y) ^(a? + ty)
14 ON CERTAIN DEFINITE INTEGRALS. [103
3. +
ft / 1 • \ _ g (a? 4 iy ) _ /(a?  f iy)
4. +, +, e
/ . • \ 7'(^+*y)
where ^, /, ^ are in fact the symbols of the inverse elliptic functions (Abel's notation)
corresponding very nearly to sin am, cos am, A am. It is remarkable that the last
value of cannot be thus expressed, but only by means of the more complicated
transcendant yx, corresponding to the H(x) of M. Jacobi. The four cases correspond
obviously to
1 . 1^ (a? + rw, y + w) = ()''+• yjr (», y),
2. ^(x^rw, y + 8v) = (y ^{x, y),
3. ^{x^rw, y + 8v) = ('y y^(x, y\
4. y}r(x + rw, y + 8v)= y^(x, y).
The above formulae may be all of them modified, as in the case of single integrals,
by means of the obvious equation
The most important particular case is
00 # 00
^ 00 #
/.J
00
(* + iy)
for in almost all the others, for example in
the second integration cannot be effected.
Suppose next ^{x, y) is one of the functions 7(a? + iy), g{x + iy), G(x\'iy\
CS (a? + iy), so that
^r(x\'rw, y + 8v) = (±y{±yUrjylr{x, y\
where
(see memoir quoted). Then, retaining the same value as before of "9 (x, y), we have
still the formula (B), in which
^ ^^ X + ly '\rw + 8m
But this summation has not yet been effected; the difficulty consists in the variable
factor €^* ('•«'«*') in the numerator, nothing being known I believe of the decomposition
of functions into series of this form.
103]
ON CERTAIN DEFINITE INTEGRALS.
15
On the subject of the preceding" paper may be consulted the following memoirs by
Raabe, "Ueber die Summation periodischer Reihen," Crelle, t. xv. [1836], pp. 355 — 364,
and •* Ueber die Summation harmonisch periodischer Reihen," t. xxiil. [1842], pp. 105 —
125, and t. xxv. [1843], pp. 160 — 168. The integrals he considers, are taken between
the limits 0, oo (instead of — oo , oo ). His results are consequently more general than
those given above, but they might be obtained by an analogous method, instead of
the much more complicated one adopted by him : thus if <^ (a? h 2'rr) = <f)x, the integral
/
90
if>x — reduces itself to
^•r*'?Ssrr'^*'[i+^"(iA™2iy •
provided I dx(t>x = 0. The summation in this formula may be effected by means of
Jo
the function F and its differential coefficient, and we have
/:
, dx
^ X
^'^)
which is in effect Raabe's formula (10), Crelle, t. xxv. p. 166.
By dividing the integral on the righthand side of the equation into two others
whose limits are 0, tt, and tt, 27r respectively, and writing in the second of these 27r — a:
instead of x, then
J ^ 29r j J
<l>x i:^h<^(27ra:)— ^
27r>'
^
'•'£)
f '>£)'
dx;
or reducing by
■ (4) '■'(' s)
— TT cot ^X,
we have
^^
j ^ — =ij <l>xcoHxdx^j [4>x h (f> (in  x)] — JT^»
^"ich corresponds to Raabe's formula (10'). If <^ ( aj) =  <f>x, so that ^ h <^ (27r — a?) = 0,
^^^ last formula is simplified ; but then the integral on the first side may be replaced
k f * dx
"Ml <l>x — ,80 that this belongs to the preceding class of formulse.
</ — ae X
16
[104
104.
ON THE THEORY OF PERMUTANTS.
[From the Cavibridge and Dublin Maihematical Journal, vol. vii. (1852), pp. 40 — 5l.]
A FORM may by considered as composed of blanks which are to be filled up by
inserting in them specializing characters, and a form the blanks of which are so filled
up becomes a symbol. We may for brevity speak of the blanks of a symbol in the
sense of the blanks of the form from which such symbol is derived. Suppose the
characters are 1, 2, 3, 4,..., the symbol may always be represented in the first
instance and without reference to the nature of the form, by F1334... And it will be
proper to consider the blanks as having an invariable order to which reference will
implicitly be made; thus, in speaking of the characters 2, 1, 3, 4,... instead of as
before 1, 2, 4,... the symbol will be V^^,., instead of V^^,.. , When the form is
given we shall have an equation such as
according to the particular nature of the form.
Consider now the characters 1, 2, 3, 4,..., and let the primitive arrangement and
every arrangement derivable from it by means of an even number of inversions or
interchanges of two characters be considered as positive, and the arrangements derived
from the primitive arrangement by an odd number of inversions or interchanges of
two characters be considered as negative ; a rule which may be termed " the rule of
signs." The aggregate of the symbols which correspond to every possible arrangement
of the characters, giving to each symbol the sign of the arrangement, may be termed
a ** Permutant ; " or, in distinction from the more general functions which will presently
be considered, a simple permutant, and may be represented by enclosing the sjTnbol
in brackets, thus {V^^,,,). And by using an expression still more elliptical than the
blanks of a symbol, we may speak of the blanks of a permutant, or the characters
of a permutant.
104] ON THE THEORY OP PERMUTANT8. 17
As an instance of a simple permutant, we may take
(r^)^r^+ v^ + r^,^ f« f« f„;
and if in particular Fia = ai6j(J», then
It follows at once that a simple permutant remains unaltered, to the sign prhs according
to the rule of signs, by any permutations of the characters entering into the per
mutant For instance,
(F„) = (F„) = (F„) =  ( F«.) =  (F„) =  (F„).
Consequently also when two or more of the characters are identical, the permutant
vanishes, thus
The form of the symbol may be such that the sjrmbol remains unaltered, to the sign
pris according to the rule of signs, for any permutations of the characters in certain
particular bUmka Such a system of blanks may be termed a quote. Thus, if the first
and second blanks are a quote,
and consequently
(F„) = 2(F„+F„+F„);
and if the blanks constitute one single quote,
( ^m . . . ) = iV K ug . . . ,
where iV=1.2.3...w, n being the number of characters. An important case, which
will be noticed in the sequel, is that in which the whole series of blanks divide
themselves into quotes, each of them containing the same number of blanks. Thus,
if the first and second blanks, and the third and fourth blanks, form quotes respectively,
It is easy now to pass to the general definition of a "Permutant." We have only
to consider the blanks as forming, not as heretofore a single set, but any number of
distinct sets, and to consider the characters in each set of blanks as permutable
inter ae and not otherwise, giving to the sjnnbol the sign compounded of the signs
corresponding to the arraDgements of the characters in the different sets of blanks.
Thus, if the first and second blanks form a set, and the third and fourth blanks form
a set,
The word 'set' will be used throughout in the above technical sense. The particular
mode in which the blanks are divided into sets may be indicated either in words or
by some superadded notation. It is clear that the theory of permutants depends
ultimately on that of simple permutants; for if in a compound permutant we first
write down all the terms which can be obtained, leaving unpermuted the characters
c. II. 3
18 ON THE THEORY OP PERMUTANT8. [104
of a particular set, and replace each of the terms so obtained by a simple permutant
having for its characters the characters of the previously unpermuted set, the result
is obviously the original compound permutant. Thus, in the abovementioned case,
where the first and second blanks form a set and the third and fourth blanks form
a set
((f;»o)=(^i~)(f.^).
in the former of which equations the first and second blanks in each of the permutants
on the second side form a set, and in the latter the third and fourth blanks in each
of the permutants on the second side form a set, the remaining blanks being simply
supernumerary and the characters in them unpermutable. It should be noted that
the term quote, as previously defined, is only applicable to a system of blanks belonging
to the same set, and it does not appear that anything would be gained by removing
this restriction.
The following rule for the expansion of a simple permutant (and which may be
at once extended to compound permutants) is obvious. Write down all the distinct
terms that can be obtained, on the supposition that the blanks group themselves in
any manner into quotes, and replace each of the terms so obtained by a compound
permutant having for a distinct set the blanks of each assumed quote; the result is
the original simple permutant. Thus in the simple permutant (Vmd, supposing for
the moment that the first and second blanks form a quote, and that the third and
fourth blanks form a quote, this leads to the equation
( F;«) = + (( F,^)) + (( r^)) + (( F,«)) + (( F^)) + (( F„,)) + (( F«„)),
where in each of the permutants on the second side the first and second blanks form
a set, and also the third and fourth blanks.
The blanks of a simple or compound permutant may of couree, without either
gain or loss of generality, be considered as having any particular arrangement in qaaoe,
for instance, in the form of a rectangle : thus F„ is neither more nor less general than
Fis4. The idea of some such arrangement naturally presents itself as affording a means
of showing in what manner the blanks are grouped into seta But, considering the
blanks as so arranged in a rectangular form, or in lines and columns, suppose in the
first instance that this arrangement is independent of the grouping of the blanks into
sets, or that the blanks of each set or of any of them are distributed at random in
the different lines and columns. Assume that the form is such that a sjnnbol
'^«^y ...
is a function of symbols Vafiy..., Va^y'...t &c. Or, passing over this general case, and
the case (of intermediate generality) of the function being a symmetrical 'function,
assume that
afi^y,..
104] ON THE THEORY OF PBRMUTANXa 19
is the product of symbols V^i^y..., V.js'y..., &c Upon this assumption it becomes
important to distinguish the different ways in which the blanks of a set are distributed
in the different lines and columns. The cases to be considered are : (A). The blanks
of a single set or of single sets are situated in more than one column. (E), The
blanks of each single set are situated in the same column. (C). The blanks of each
single set form a separate column. The case (B) (which includes the case (C)) and the
case (C) merit particular consideration. In fact the case (E) is that of the functions
which I have, in my memoir on Linear Transformations in the Journal, [13, 14]
called hyperdeterminants, and the case ((7) is that of the particular class of hyper
determinants previously treated of by me in the Cambridge PhUoaophical Tranaactiona,
[12] and also particularly noticed in the memoir on Linear Transformations. The
functions of the case (B) I now propose to call '' Intermutants," and those in the case
(C) *' Commutants." Commutants include as a particular case '' Determinants," which
term will be used in its ordinary signification. The case (A) I shall not at present
discuss in its generality, but only with the further assumption that the blanks form a
single set (this, if nothing further were added, would render the arrangement of the
blanks into lines and columns valueless), and moreover that the blanks of each line
form a quote: the permutants of this class (from their connexion with the researches
of Pfaflf on differential equations) I shall term "Pfaffians." And first of commutants,
which, as before remarked, include determinants.
The general expression of a commutant is
(^11 ); or ai ...^
11...
22
nn
22
nn
J
and (stating again for this particular case the general rule for the formation of a
permutant) if, permuting the characters in the same column in every possible way,
considering these permutations as positive or negative according to the rule of signs,
one system be represented by
'1 " 1 • • •
the commutant is the sum of all the different terms
The different permutations may be formed as follows: first permute the characters in
all the columns except a single column, and in each of the arrangements so obtained
permute entire lines of characters. It is obvious that, considering any one of the
arrangements obtained by permutations of the characters in all the columns but one,
the permutations of entire lines and the addition of the proper sign will only reproduce
3—2
20
ON THE THEORY OP PERMUTANT8.
[104
the same 83nnbol — in the case of an even number of columns constantly with the
positive sign, but in the case of an odd number of columns with the positive or
negative sign, according to the rule of signs. For the inversion or interchange of two
entire lines is equivalent to as many inversions or interchanges of two characters as
there are characters in a line, that is, as there are columns, and consequently intro
duces a sign compounded of as many negative signs as there are columns. Hence
Theoreh a commutant of an even number of columns may be calculated by
considering the characters of any one column (no matter which) as supernumerary
unpermutable characters, and multiplying the result by the number of permutations of
as many things as there are lines in the commutant
The mark f* added to a commutant of an even number of columns will be employed
to show that the numerical multiplier is to be omitted. The same mark placed over
any one of the columns of the commutant will show that the characters of that
particular column are considered as nonpermutable.
A determinant is consequently represented indiflferently by the notations
11^
t
t
>
+
ir
22
•
22
•
22
•
•
•
, KM ,
•
and a commutant of an odd number of cohimns vanishes identically.
By considering, however, a commutant of an odd number of columns, having the
characters of some one column nonpermutable, we obtain what will in the sequel be
Hpoken of as commutants of an odd number of columns. This nonpermutability will be
denoted, as before, by means of the mark f placed over the column in question, and
it is to be noticed that it is not, as in the case of a commutant of an even number
of columns, indifferert over which of the columns the mark in question is placed; and
consequently there would be no meaning in simply adding the mark f to a com
mutant of an odd number of columns.
A commutant is said to be symmetrical when the symbols Fo/jy... are such as to
remain unaltered by any permutations inter se of the characters a, /8, 7 . . . A com
mutant is said to be skew when each symbol V^py, is such as to be altered in sign
only according to the rule of signs for any permutations inter se of the characters
a, /9, 7 . . . , this of course implies that the symbol Va^y.,. vanishes when any two of
the characters a, 13, 7... are identical. The commutant is said to be demiskew when
Fa,^.y... is altered in sign only, according to the rule of signs for any permutation
inter se of nonidentical characters a, /8, 7,...
An intermutant is represented by a notation similar to that of a commutant. The
sets are to be distinguished, whenever it is possible to do so, by placing in contiguity
the symbols of the same set, and separating them by a stroke or bar from the symbols
104]
ON THE THEORY OP PERMUTANT8.
21
of the adjacent sets. If, however, the symbols of the same set cannot be placed con
tiguously, we may distinguish the symbols of a set by annexing to them some auxiliary
character by way of su£Bx or otherwise, these auxiliary symbols being omitted in the
final result. Thus
ri 1 la)
2
3
4
2
3
3
2b
oa
6b
would show that 1, 2 of the first column and the 3, 4 of the same column, the 1, 2
and the upper 3 of the second column, and the lower 3 of the same column, the 1, 5
of the third column, and the 2, 6 of the same column, form so many distinct sets, —
the intermutant containing therefore
(2.2.6.1.2.2 = ) 96 terms.
A commutant of an even number of columns may be considered as an intermutant
such that the characters of some one (no matter which) of its columns form each of
them by itself a distinct set, and in like manner a commutant of an odd number of
columns may be considered as an intermutant such that the characters of some one
determinate column form each of them by itself a distinct set
The distinction of sjnmmetrical, skew and demiskew applies obviously as well to
intermutants as to commutants. The theory of skew intermutants and skew commutants
has a connexion with that of Pfa£Bans.
Suppose F.^y... = V^+fi+y... (which implies the sjrmmetry of the intermutant or com
mutant) and write for shortness F© = a, Fi = 6, Fj = c, &c. Then
\i
1
1
T
[j J]=(acn &C.
The functions on the second side are evidently hyperdeterminants such as are
discussed in my memoir on Linear Transformations, and there is no diflSculty in
forming directly from the intermutant or commutant on the first side of the equation
the symbol of derivation (in the sense of the memoir on Linear Transformations) from
which the hyperdeterminant is obtained. Thus
ri'
is 12 . UU,
r]'
is liUoU',
1
1
1
1
is 12 .UU.
12U'U\
22
ON THE THEORY OF PEBMUTANTS.
[104
each pennutable column corresponding to a 12(') and a nonpermutable column
1 1
changing JJV into U'U'K Similarly
CO 0^
1 1
l2 2)
becomes (12 . 13 . 23)* . UUU,
ro
t
0]
1
1
.2
2)
becomes 12. 13.28 U'*V^^U\
ro
0]
1
1
2
2
3
V
sj
zr— _— . S
becomes (12 . IS . 14 . 23 . 24 . 84) UUUU, &c
The analogy would be closer if in the memoir on Linear Transformations, just as
12 is used to signify
, 123 had been used to signify
kic,, for
then
ro
0]
1
1
u
2)
would have corresponded to 123 .UUU,
ro
0]
1
1
u
2}
to 128 £r»[7'I7»; and this
would not only have been an addition of some importance to the theoij*, but would
in some instances have facilitated the calculation of hyperdeterminants. The preceding
remarks show that the intermutant
ro 0^
1 1 T
U 1 ^J
(where the first and fourth blanks in the last column are to be considered as belonging
to the same set) is in the hyperdeterminant notation (12 . 34)*.(14. 23) [7Z7[7£r.
1 Viz. corresponds to l2 beoaase and 1 are the characters ooonpying the first and second blanks of a oolomn.
1
If and 1 had been the characters occapying the second and third blanks in a column, the symbol would have been
23 and so on. It will be remembered, that the symbolic nombers 1, 2 in the hyperdeterminant notation are
merely introdaoed to distinguish from each other functions which are made identical after certain differentiations
are performed.
104]
ON THE THEORY OF PERMUTANTS.
?3
It will, I think, illustrate the general theory to perform the development of the
lastmentioned intermntant. We have
'0 0^
as
+
0'
—
t
0'
—
t
'0 r
+
t
^0 1^
111
1 I 1
110
1 1 1
1 1
1
1
1
ll 1 IJ
a 1 1.
.111
A 1 0.
.1 1 0>
^2 (TO on ro oiro o] ro o it
iLi 1 iJ Ll 1 iJ Ll 1 oj Ll 1 iJ
= 2{(ad6c)«4(ac6»)(Wc»)},
= 2 (a»(? + 4ac» + 46»rf  36»c»  6abcd),
the different steps of which may he easily verified.
The following important theorem (which is, I believe, the same as a theorem of
Mr Sylvester's, published in the Philosophical Magazine) is perhaps best exhibited by
means of a simple example. Consider the intermutant
'x
r
y
4
X
3
u
2.
where in the first column the sets are distinguished as before by the horizontal bar,
but in the second column the 1, 2 are to be considered as forming a set, and the
3, 4 as forming a second set. Then, partially expanding, the intermutant is
'x
r
—
'y
V
—
'x
V
+
'y
r
y
4
X
4
y
4
X
4
X
3
X
3
y
3
y
3
y
2.
y
2.
<x
2>
.X
2.
or, since entire horizontal lines may obviously be permuted,
+
'x V
—
t
'y
r
•^
+
'x
I'
+
t
> 1^
y 2
y
2
X
2
X 2
X 3
X
3
y
3
y 3
^X 4;
.y 4>
.«
4>
.y
*.
24
ON THE THEORY OF PERMUTANTS.
[104
and, observing that the 1, 2 form a permutable system as do also the 3, 4, the
second and third terms vanish, while the first and fourth terms are equivalent to
each other; we may therefore write
'x
V
s
'x
V
y
2
y
4
X
3
X
3
.y
4>
.y
2.
where on the first side of the equation the bar has been introduced into the second
column, in order to show that throughout the equation the 1, 2 and the 3, 4 are
to be considered as forming distinct sets.
Consider in like manner the expression
(x U
y
z
7
6
X 8
y 2
z 9
X 4
y 5
^z Sj
where in the first column the sets are distinguished by the horizontal bars and in
the second column the characters 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be
considered as belonging to distinct sets. The same reasoning as in the former case
will show that this is a multiple of
X
V
y
2
z
3
X
"i
y
5
z
6
X
7
y
8
J
9.
and to find the numerical multiplier it is only necessary to inquire in how many
wajTS, in the expression first written down, the characters of the first column can be
104]
ON THB THEORY OF PBRMUTANTS.
25
permuted so that x, y, z may go with 1, 2, 3 and with 4s, 5, 6 and with 7, 8, 9.
The order of the x, y, z ia the second triad may be considered as arbitrary; but
onoe assumed, it determines the place of one of the letters in the first triad; for
instance, xS and z9 determine y7. The first triad must therefore contain xl and z6
or x6 and zl. Suppose the former, then the third triad must contain zS, but the
remaining two combinations may be either x4i, y5, or x5, y4. Similarly, if the first
triad contained x6, zl, there would be two forms of the third triad, or a given
form of the second triad gives four different forms. There are therefore in all
24 forms, or
t
24
'x V
=
'x r
y 2
y 7
z 3
X 4
z 6
X 8
y 5
y 2
z 6
X 7
z 9
X 4
y 8
y 5
.^ 9.
z 3
where the bars in the second column on the first side show that throughout the
equation 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be considered as forming distinct
sets. The above proof is in reality perfectly general, and it seems hardly necessary
to render it so in terms.
To perceive the significance of the above equation it should be noticed that the
first side is a product of determinants, viz.
24
'x
r
t
'x
6^
t
'x
7'
y
2
y
6
y
8
.z
3.
.z
7.
.z
9.
t;
and if the second side be partially expanded by permuting the characters of the
second column, each of the terms so obtained is in like manner a product of deter
minants, so that
24
/
X
r
t
'x
4^
t
'x
T
t =
'x
V
t
'x
8^
t
'x
4^
y
2
y
5
y
8
y
7
y
2
y
5
.e
3.
.e
6.
.»
9>
.z
6.
.«
9.
.«
3.
+ ±&c.,
the permutations on the second side being the permutations inter se of 1, 2, 3, of
4, 5, 6, and of 7, 8, 9.
It is obvious that the preceding theorem is not confined to intermutants of two
columns.
c. n. 4
26 ON THE THEORY OF PBRMUTANTS. [104
POSTSCRIPT.
I wish to explain as accurately as I am able, the extent of my obligations to Mr Sylvester in
respect of the subject of the present memoir. The term permutant is due to him — ^intemiutant and
commutant are merely terms framed between us in analogy with permutant, and the names date from
the present year (1851). The theory of commutants is given in my memoir in the Cambridge PhUo
iophical Transactionij [12], and is presupposed in the memoir on Linear Transformations, [13, 14]. It
will appear by the lastmentioned memoir that it was by representing the coefficients of a biquadratic
function by a = 1111, 6 = 1112 = 1121 «=&c, c=1122s&c., c?=1222 = &c., 6 = 2222, and forming the
commutant Mlll^ that I was led to the function (w4W+3c*. The function aoe+ibcdcuPhei"^
[ 2222 J
or a, 6, c is mentioned in the memoir on Linear Transformations, as brought into notice by
bf Cj d
c, dy e
Mr Boole. From the particular mode in which the coefficients a, 6,... were represented by symUdH
such as 1111, &c., 1 did not perceive that the lastmentioned function could be expressed in the
commutant notation. The notion of a permutant, in its most general sense, is explained by me in
my memoir, "Sur les determinants gauches,'' Ordley t xzzvii. pp. 93 — 96, [69]; see the paragraph
(p. 94) commencing " On obtient ces fonctions, &c,** and which should run as follows : *' On obtieiit
oes fonctions (dont je reprends ici la thdorie) par les propriety gdnerales d'un determinant d^fini
comme suit. En exprimant &c. ;" the sentence as printed being " d^fini. Car en exprimant &c.,''
which confuses the sense. [The paragraph is printed correctly 69, p. 411.] Some time in the presieut
year (1851) Mr Sylvester, in conversation, made to me the very important remark, that as one of a
class the abovementioned function,
aoe + ibcd ad^^l^c^,
could be expressed in the commutant notation ( ^ , viz. by considering 00 = a, 01 «» 10 = />,
1 1
.2 2;
02 = llai20=c, 12»21=cf, 22 =e; and the subject being thereby recalled to my notice, I found
shortly afterwards the expression for the function
a W + 4ac» + 463rf  36M  Qabod
(which cannot be expressed as a commutant) in the form of an intermutant, and 1 was thence leii
to see the identity, so to say, of the theory of hyperdeterminants, as given in the memoir on
Linear Transformations, with the present theory of intermutants. It is understood between Mr Sylvester
and myself, that the publication of the present memoir is not to affect Mr Sylvester's right to
claim the origination, and to be considered as the first publisher of such part as may belong to him
of the theory hero sketched out.
105]
27
105.
CORRECTION OF THE POSTSCRIPT TO THE PAPER ON
PERMUTANTS.
[From the Cambridge and Dublin MathenuUical Journal, vol. vii. (1852), pp. 97 — 98.]
Mr Sylvester has represented to me that the paragraph relating to his com
munications conveys an erroneous idea of the nature, purport, and extent of such
communications; I have, in fact, in the paragraph in question, singled out what imme
diately suggested to me the expression of the function Qabcd + 36*0* — 400* — 46*^ — a'd*
as a partial commutant or intermutant, but I agree that a fuller reference ought to
have been made to Mr Sylvester's results, and that the paragraph in question would
more properly have stood as follows:
'* Under these circumstances Mr Sylvester communicated to me a series of formal statements,
not only oral but in writing, to the effect that he had discovered a permutation method of obtaining
SLA many invariants — viz. commutantive invariants — by direct inspection from a function of any degree
of any nimiber of letters as the index of the degree contains even factors ; one of these conunu
tantive invariants being in fact the function ace^^hcdae^bd^i^^ expressible, according to Mr
fa^ ah 6*\
Sylvester's notation, l>y ( 2' ^ jsl)'* ^^d, according to the notation of my memoir in the Camb.
PhU. Tram., supposing 00= a, 01 = 10=6, 02 = ll = 20=c, &c. by
00
11
22
»
Mr Sylvester and I shall, I have no doubt, be able to agree to a joint statement
i>f any further correction or explanation which may be required.
4—2
28
[106
106.
ON THE SINGULAKITIES OF SUEFACES.
[From the Cambridge and Dublin Mathematical Journal, vol. vii. (1852), pp. 166 — 171.]
In the following paper, for symmetry of nomenclature and in order to avoid
ambiguities, I shall, with reference to plane curves and in various phrases and
compoimd words, use the term •*node" as synonymous with double point, and the
term "spinode" as synonymous with cusp. I shall, besides, have occasion to consider
the several singularities which I call the "flecnode," the "oscnode," the "fleflecnode,"
and the "tacnode:" the flecnode is a double point which is a point of inflexion on
one of the branches through it; the oscnode is a double point which is a point of
osculation on one of the branches through it; the fleflecnode is a double point which
is a point of inflexion on each of the branches through it; and the tacnode is a
double point where two branches touch. And it may be proper to remark here, that
a tacnode may be considered as a point resulting from the coincidence and amalga
mation of two double points (and therefore equivalent to twelve points of inflexion);
or, in a different point of view, [?] as a point uniting the characters of a spinode and
a flecnode. I wish to call to mind here the definition of conjugate tangent lines of
a surfiEu;e, viz. that a tangent to the curve of contact of the surface with any
circumscribed developable and the corresponding generating line of the developable,
are conjugate tangents of the surface.
Suppose, now, that an absolutely arbitrary surface of any order be intersected
by a plane: the curve of intersection has not in general any singularities other than
such as occur in a perfectly arbitrary curve of the same order; but as a plane can
be made to satisfy one, two, or three conditions, the curve may be made to acquire
singularities which do not occur in such absolutely arbitrary curve.
Let a single condition only be imposed on the plane. We may suppose that
the curve of intersection has a node; the plane is then a tangent plane and the
node is the point of contact — of course any point on the surface may be taken for
106] ON THE SINGULARITIES OF SURFACES. 29
the node. We may if we please use the term "nodes of a surface," "nodeplanes of
a surface/' as synonymous with the points and tangent planes of a surface. And it
will be convenient also to use the word nodetangents to denote the tangents to the
curve of intersection at the node; it may be noticed here that the nodetangents
are conjugate tangents of the surface.
Next let two conditions be imposed upon the plane: there are three distinct
cases to be considered.
First, the curve of intersection may have a flecnode. The plane (which is of
course still a tangent plane at the flecnode) may be termed a flecnodeplane ; the
flecnodes are singular points on the surface lying on a curve which may be termed
the " flecnodecurve ^" and the flecnodeplanes give rise to a developable which may
be termed the flecnodedevelope. The " flecnodetangents " are the tangents to the
curve of intersection at the flecnode; the tangent to the inflected branch may be
termed the "singular flecnodetangent," and the tangent to the other branch the
"ordinary flecnodetangent."
Secondly, the curve of intersection may have a spinode. The plane (which is of
course still a tangent plane at the spinode) may be termed a spinodeplane ; the
spinodes are singular points on the surface lying on a curve which may be termed
the "spinodecurve*." And the spinodeplanes give rise to a developable which may
be termed the " spinodedevelope." Also the " spinodetangent " is the tangent to the
curve of intersection at the spinode.
Thirdly, the curve of intersection may have two nodes, or what may be termed
a "nodecouple." The plane (which is a tangent plane at each of the nodes and
therefore a double tangent plane) may be also termed a "nodecoupleplane." The
nodecouples are pairs of singular points on the surface lying in a curve which may
be termed the "nodecouplecurve," and the nodecoupleplanes give rise to a deve
lopable which may be termed the " nodecoupledevelope." The tangents to the curve
of intersection at the two nodes of a nodecouple might, if the term were required,
be termed the "nodecoupletangents." Also one of the nodes of a nodecouple may
be termed a " nodewithnode," and the tangents to the curve of intersection at such
point will be the " nodewithnodetangents."
1 The fleonodeonrve, defined as the looas of the points through which can be drawn a line meeting the surface
in four conseoaftive points, was, so far as I am aware, first noticed in Mr Salmon's paper **0n the Triple
Tangent Planet of a Surface of the Third Order'* {Journal, t. iv. [1849], pp. 252—260), where Mr Salmon,
unong other things, shows that the order of the surface being fi, the curve in question is the intersection of
the surface with a surface of the order lln24.
* The notion of a spinode, considered as the point where the indicatriz is a parabola (on which account
the spinode has been termed a parabolic point) may be found in Dupin's Diveloppements de OiomStrie : the
most important step in the theory of these points is contained in Hesse's memoir "Ueber die Wendepuncte
der Curren dritter Ordnnng" {CreUe, t. xxvin. [1848], pp. 97 — 107), where it is shown that the spinodecurve
is the curre of intersection of the surface supposed as before of the order n, with a certain surface of the
order 4(fi2). See also Mr Salmon's memoir "On the Condition that a Plane should touch a surface along
a Corre Line" {Jovrnal, t. in. [1848], pp. 44—46).
30 ON THE SINGULARITIES OF SURFACES. [l06
It is hardly necessary to remark that the flecnodecurve is not the edge of
regression of the fleenodedevelope, and the like remark applies m,fn. to the spinode
curve and the nodecouple curve.
Finally, let three conditions be imposed upon the plane: there are six distinct
cases to be considered, in each of which we have no longer curves and developes,
but only singular points and singular tangent planes determinate in number.
First, the curve of intersection may have an oscnode. The plane (which continues
a tangent plane at the oscnode) is an " oscnedeplane.'' The '' oscnodetangents " are
the tangents to the curve of intersection at the oscnode ; the tangent to the
osculating branch is the " singular oscnodetangent ; " and the tangent to the other
branch the "ordinary oscnodetangent."
Secondly, the curve of intersection may have a fleflecnode. The plane (which
continues a tangent plane at the fleflecnode) is a " fleflecnodeplane." The " fleflec
nodetangents " are the tangents to the curve of intersection at the fleflecnode.
Thirdly, the curve of intersection may have a tacnode. The plane (which
continues a tangent plane at the tacnode) is a " tacnodeplane.'' The ''tacnode
tangent" is the tangent to the curve of intersection at the tacnode.
Fourthly, the curve of intersection may have a node and a flecnode, or what
may be termed a nodeandflecnode. The plane (which is a tangent plane at the
node and also at the flecnode, where it is obviously a flecnodeplane) is a "nodeand
flecnodeplane." The " nodeandflecnodetangents," if the term were required, would be
the tangents to the curve of intersection at the node and at the flecnode of the
uodeandflecnode. The node of the nodeandflecnode may be distinguished as the
nodewithflecnode, and the flecnode as the flecnodewithnode, and we have thus the
terms " nodewithflecnodetangents," " flecnodewithnodetangents," " singular flecnode
withnodetangent," and "ordinary flecnode withnodetangent."
Fifthly, the curve of intersection may have a node and also a spinode, or what
may be termed a " nodeandspinode." The plane (which is a tangent plane at the
node, and is also a tangent plane at the spinode, where it is obviously a spinodeplane)
is a " nodeandspinodeplane." The nodeandspinodetangents, if the term were
required, would be the tangents at the node and the tangent at the spinode of the
nodeandspinode to the curve of intersection. The node of the nodeandspinode
may be distinguished as the " nodowithspinode," and the spinode as the "spinode
Avithnode," and we have thus the terms " nodewithspinodetangent," " spinodewithnode
tangent."
Sixthly, the curve of intersection may have three nodes, or what may be termed
a "nodetriplet." The plane (which is a triple tangent plane touching the surfieu^ at
each of the nodes) is a "nodetripletplane." The "nodetriplettangents," if the term
were required, would be the tangents to the curve of intersection at the nodes of
the nodetriplet. Each node of the nodetriplet may be distinguished as a "node
106 J ON THE SINGULARITIES OP SURFACES. 31
withnodecouple," and the tangents to the curve of intersection at such nodes are
•'nodewithnodecoupletangents." The terms " nodecouplewithnode," '' nodecouplewith
nodetangent," might be made use of if necessary.
It should be remarked that the oscnodes lie on the flecnodecurve, as do also
the fleflecnodes; these latter points are real double points of the flecnodecurve. The
tacnodes are points of intersection and (what will appear in the sequel) points of
contact of the flecnodecurve, the spinodecurve, and the nodecouplecurve. The spinode
withnodes are points of intersection of the spinodecurve and nodecouplecurve, and
the flecnodewithnodes are points of intersection of the flecnodecurve and nodecouple
curve; the nodewithnodecouples are real double points (entering in triplets) of the
nodeoouplecurve.
Consider for a moment an arbitrary curve on the surface; the locus of the node
tangents at the different points of this curve is in general a skew surface, which
may however, in cases to be presently considered, degenerate in different ways.
Reverting now to the flecnodecurve, it may be shown that the singular flecnode
tangent coincides with the tangent of the flecnodecurve. For consider on a surface
two consecutive points such that the line joining them meets the surfistce in two
points consecutive to the firstmentioned two points. The line meets the surface in
four consecutive points, it is therefore a singular flecnodetangent ; each of the first
mentioned two points must be on the flecnodecurve, or the singular flecnodetangent
touches the flecnodecurve. The two flecnodetangents are by a preceding observation
conjugate tangents. It follows that the skew surface, locus of the flecnodetangents,
brides up into two surfaces, each of which is a developable, viz. the locus of the
singular flecnodetangents is the developable having the flecnodecurve for its edge of
regression, and the locus of the ordinary flecnodetangents is the flecnodedevelope.
Of course at the tacnode, the tacnodetangent touches the flecnodecurve.
Passing next to the spinodecurve, the spinodeplane and the tangentplane at a
consecutive point along the spinodetangent are identical^ or their line of intersection
is indeterminate. The spinodetangent is therefore the conjugate tangent to any other
tangent line at the spinode, and therefore to the tangent to the spinodecurve. It
follows that the surfEtce locus of the spinodetangents degenerates into a developable
sar£BU» twice repeated, viz. the spinodedevelope. Consider the tacnode as two coin
cident nodes; each of these nodes, by virtue of its constituting, in conjunction with
the other, a tacnode, is on the spinodecurve; or, in other words, the tacnodetangent
touches the spinodecurve, and the same reasoning proves that it touches the node
couplecurva It has already been seen that the tacnodetangent touches the flecnode
curve ; consequently the tacnode is a point, not of simple intersection only, but of
omtact, of the flecnodecurve, the spinodecurve, and the nodecouplecurve.
In virtue of the principle of the spinodeplane being identical with the tangent
plane at a consecutive point along the spinode tangent, it appears that the tacnode
1 It most not be inferred that the tangent plane at rach oonBeeotiTe point is a spmodeplane ; thia ii
obfTioody not the ease.
32 ON THE SINGULARITIES OF SURFACES. [106
plane is a stationary plane, as well of the flecnodedevelope as of the spinode
develope, and it would at first sight appear that it must be also a stationary
tangent plane of the nodecoupledevelope. But this is not so; the nodewithnode
planes envelope, not the nodecoupledevelope, but the nodecoupledevelope twice
repeated: the tacnodeplane is in a sense a stationary plane on such duplicate
developable, but not in any manner on the single developable. The tacnodeplane is
an ordinary tangent plane of the nodecoupledevelope.
Consider now a spinodewithnode, which we have seen is a point of intersection
of the spinodecurve and nodecouplecurve. The tangent plane at a consecutive point
along the spinodewithnodetangent, is identical with the spinodewithnodeplane ; the
curve of intersection of the tangent plane at such consecutive point has therefore a
node at the nodewithspinode, or the tangent plane in question is a nodecouple
plane, and the point of contact is a point on the nodecouplecurve. Consequently
the spinodewithnodetangent touches the nodecouplecurve, and thence also the
spinodewithnodeplane is a stationary tangent plane of the nodecoupledevelope.
It should be remarked that no circumscribed developable can have a stationary
tangent plane except the tangent planes at the points where the curve of contact
meets the spinodecurve, and any one of these planes is only a stationary plane
when the curve of contact touches the spinodetangent ; and that the nodecouple
curve and the flecnodecurve do not intersect the spinodecurve except in the points
which have been discussed
Recapitulating, the nodecouplecurve and the spinodecurve touch at the tacnodes,
and intersect at the spinodewithnodes : moreover, the tacnodeplanes are stationary
planes of the spinodedevelope, and the spinodewithnodeplanes are stationary planes
of the nodecoupledevelope. Besides this, the two curves are touched at the tacnodes
by the flecnodecurve, and the tacnodeplanes are stationary planes of the flecnode
develope.
107]
33
107.
ON THE THEORY OF SKEW SURFACES.
[From the Cambridge and Dublin Mathematical Journal, vol. vii. (1862), pp. 171 — 173.]
A SURFACE of the n^ order is a surface which is met by an indeterminate line
in n points. It follovrs immediately that a surface of the n^ order is met by an
indeterminate plane in a curve of the n^ order.
Consider a skew sur&ce or the surface generated by a singly infinite series of
lines, and let the surfeice be of the n^ order. Any plane through a generating line
meets the sur£Gtce in the line itself and in a curve of the (n — 1)^ order. The
generating line meets this curve in (n — 1) points. Of these points one, viz. that
adjacent to the intersection of the plane with the consecutive generating line, is a
unique point ; the other (n — 2) points form a systenL Each of the (n — 1) points
are svb modo points of contact of the plane with the surface, but the proper point
of contact is the unique point adjacent to the intersection of the plane with the
consecutive generating line. Thus every plane through a generating line is an ordinary
tangent plane, the point of contact being a point on the generating line. It is not
necessary for the present purpose, but I may stop for a moment to refer to the
known theorems that the anharmonic ratio of any four tangent planes through the
same generating line is equal to the anharmonic ratio of their points of contact, and
that the locus of the normals to the sur&ce along a generating line is a hyperbolic
paraboloid. Returning to the (n — 2) points in which, together with the point of
contact, a generating line meets the curve of intersection of the sur&ce and a plane
through the generating line, these are fixed points independent of the particular plane,
and are the points in which the generating line is intersected by other generating
lines. There is therefore on the surfSEU^ a double curve intersected in (n — 2) points
by each generating line of the surfSsu^e — a property which, though insufficient to
determine the order of this double curve, shows that the order cannot be less than
(r  2X (Thus for n = 4, the above reasoning shows that the doublecurve must be
an. 5
34 ON THE THEORY OF SKEW SURFACES. [l07
at least of the second order: assuming for a moment that it is in any case precisely
of this order, it obviously cannot be a plane curve, and must therefore be two non
intersecting lines. This suggests at any rate the existence of a class of skew surfaces
of the fourth order generated by a line which always passes through two fixed lines
and by some other condition not yet ascertained; and it would appear that surfaces
of the second order constitute a degenerate species belonging to the class in question.)
In particular cases a generating line will be intersected by the consecutive
generating line. Such a generating line touches the double curve.
Consider now a point not on the surface ; the planes determined by this point
and the generating lines of the surface are the tangent planes through the point;
the intersections of consecutive tangent planes are the tangent lines through the
point; and the cone generated by these tangent lines or enveloped by the tangent
planes is the tangent cone corresponding to the point. This cone is of the n^ class.
For considering a line through the point, this line meets the surface in n points,
i.e. it meets n generating lines of the surface; and the planes through the line and
these n generating lines, are of course tangent planes to the cone : that is, n tangent
planes can be drawn to the cone through a given line passing through the vertex.
The cone has not in general any lines of inflexion, or, what is the same thing,
stationary tangent planes. For a stationary tangent plane would imply the inter
section of two consecutive generating lines of the surface. And since the number of
generating lines intersected by a consecutive generating line, and therefore the number
of planes through two consecutive generating lines, is finite, no such plane passes
through an indeterminate point. The tangent cone will have in general a certain
number of double tangent planes; let this number be x. We have therefore a cone
of the class n, number of double tangent planes x, number of stationary tangent
planes 0. Hence, if m be the order of the cone, a the number of its double lines,
and fi the number of its cuspidal or stationary lines,
7M = n (n — 1) — 2a?,
/3 = 3w (w  2)  6x,
a = in (n  2) (n»  9)  2« (w»  n  6) + 2« (a:  1).
This is the proper tangent cone, but the cone through the double curve is sub modo
a tangent cone, and enters as a square factor into the equation of the general
tangent cone of the order n (n — 1). Hence, if X be the order of the double curve,
and therefore of the cone through this curve,
m » 2X = n (ti  1), and therefore X = x\
that is, the number of double tangent planes to the tangent cone is equal to the
order of the double curve. It does not appear that there is anything to determine
x\ and if this is so, skew surfaces of the v!^ order may be considered as forming
dififerent families according to the order of the double curve upon them.
To complete the theory, it should be added that a plane intersects the surface
in a curve of the n*** order having x double points but no cusps.
108]
35
108.
ON CERTAIN MULTIPLE INTEGRALS CONNECTED WITH THE
THEORY OF ATTRACTIONS.
[From the Cambridge and Dublin Mathematicai Journal, vol. vii. (1852), pp. 174 — 178.]
It is easy to deduce from Mr Boole's formula, given in my paper " On a Multiple
Integral connected with the theory of Attractions," Journal, t. ii. [1847], pp. 219 — 223,
[44], the equation
df dv .^ f9:it^ r «»' {0x* afds
^/{(S($)■!
where n is the number of variables of the multiple integral, and the condition of the
integration is
(g «.)•, (1?  ft y , =,.
also where
and e is the positive root of
, (gg.)' . (8$,y ^ir
0r jr + ^••■+7
€ + — e+ —
Suppose /= jr... = ^i, and write (a — o,)' + ... = i', we obtain
ii(?«)*+«^]»"« f{inq)r(q+ljj . (l+«)*» •
5—2
36 ON CERTAIN MULTIPLE INTEGRALS [108
the limiting condition for the multiple integral being
and the function a, and limit e, being given by
€ denoting, as before, the positive root. Observing that the quantity under the integral
sign on the second side vanishes for « = e, there is no difficulty in deducing, by a
differentiation with respect to Ou the formula
i[(f«)'...+t^]*^"T(ing)r(g)j.
where (2S is the element of the surface (^^1)' + ... = ^i', and the integration is
extended over the entire surfieu^e,
A slight change of form is convenient We have
if we suppose
The formulae then become
r df... TT** r (gV h x^  i;')g (fa
i[(f'«)" + «^F"^ r(in9)r(9 + l)J. «(! + «)**+« •
in which e is the positive root of the equation
I propose to transform these formula by means of the theory of images ; it will be con
venient to investigate some preliminary formulae. Suppose X* = o" + /S" . . . , V = «i* + A*. . . ;
also consider the new constants a, 6,..., Oi, 61,..., m, /i, determined by the equations
where S is arbitrary. Then, putting
108] CONNECTED WITH THE THEORY OF ATTRACTIONS. 37
it is easy to see that
= V
Proceeding to express the single integrals in terms of the new constants, we have in
the first place A" = 8*4", where
or if we write
ooi + hhi ... = ZZi oos (k>,
we have
Hence also x~^3» where
,•  A* jfc. "'
whence
._ 1 1 2%co8a>
where p* = P + ii* — 2ZZi cos o), that is
consequently ^jV + x* "" *^ = ^*n, where TI is given by
n^ f' :^ iP'^^'A')
8 —
and it is clear that e will be the positive root of
It may be noticed that, in the particular case of ti = 0, the roots of this equation
are 0, and — — J}^ '^^. Consequently if f^—fi^ and li^fi are of opposite signs,
we have 6 = 0; but \t ff^^ and ii'/i» are of the same sign, €=^^ •'7^'— ^
38 ON CERTAIN MULTIPLE INTEGRAX8 [108
In order to transform the double integrals, considering the new variables x, y, ...,
I write 45* + y'... =r" and
whence also, if f' + i7*+ ... =p* (which gives rp = S^), we have
a: = — —
. I • • * 9
also it is immediately seen tli*t
(fa)»+ ... «•= (ft ^^^.) ^ K^  «)• + ••• +«*}.
(f  a,)» ... ^,. = ^^^^{(a,a,)»+ ... /.'} :
and from the latter equation it follows that the limiting condition for the first integral
is (^— a,)'+... >/i' (there is no difficulty in seeing that the sign < in the former
limiting condition gives rise here to the sign >), and that the second integral has to be
extended over the surface (^ — 01)*+ ... =/i*. Also if dS represent the element of this
surface, we may obtain
did'n.,.^^dxdy ,.,, dX = ^^dS;
and, combining the above formulae, we obtain
r dxdy ...
J (a^ + y« ... )*'»+« {(x  a)«f (y 6)' ... + i/'j*~«
"r(i^9)r(j+i)(?+w«)*'»^j. «(i+«)^+^'
the limiting condition of the multiple integral being
(^a,)' + (y6i)«...5/,«;
and
[ dS
~ r ( jn  9) r? (i« + 1*«)*'^ (/i> /i>) J . (1 + «)***^^* '
where d^f is the element of the surface (a?— ai)* + (y — Ji)" ... =/i', and the integration
extends over the entire surface. In these formute, I, li, p, 11 denote as follows:
p = a«+6»+..., /i' = ai» + 6i«+..., p» = (aai)> + (66x)»+ ... ,
and € is the positive root of the equation 11 = 0.
108] CONNECTED WITH THE THEORY OF ATTRACTIONS. 39
The only obviously integrable case is that for which in the second fonnula q = l\
this gives
/
dS 2w^f,
(a^ + y» ...)*~ {(a? a)> + (y 6)« + tA>}*'*» r(iw)(P + t4»)^»(Zi»/i')(l +€)*~' '
In the case of t* = 0, we have, as before, when p*— /i* and l^—fi are of opposite
signs, € = 0, and therefore 1 f € = 1 ; but when p* — /i* and i,' — /i' are of the same
sign, the value before found for 6 gives
1 + « = ^. i^/' + (p* />') (^' /.*)}•
Consider the image of the origin with respect to the sphere (a? — 01)*+ (y — 6i)"..^=/i',
the coordinates of this image are
> ••• »
and consequently, if /a be the distance of this image from the point (a, 6 ...), we have
/*'= {a ^,(i.' /.*)}»+.. .
= ^ (^'/.' + (I'' /x')W /.')};
whence, by a simple reduction,
or the values of the integral are
/)•/,» and k'/i* opposite signs, ^ = f^^ F^^fc/O'
n— 1
p* /.' and /,' /,• the same sign, / = ^?^^ ___^^__ ,
where fi is the distance from the point (a, 6...) of the image of the origin with respect
to the sphere (x — a^y + . . . — /i' = 0.
Stone Buildinga, AvguM 6, 1850.
40
[109
109.
ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL
EQUATIONS.
[From the Cambridge and Dublin Mathematical Journal, vol. viii. (1853), pp. 97 — 101]
Suppose
^ + y = 0, ic» = a, y* = 6 ;
then if we multiply the first equation by 1, j^, and reduce by the two others, we havt^
from which, eliminating x, y.
x+ y
= 0.
bx + ay = 0,
I, 1
= C
b, a
which is the equation between a and 6; or, considering x, y sa quadratic radicals,
the rational equation between x, y. So if the original equation be multiplied by a, //,
we have
a + xy0,
b + xy^O;
or, eliminating 1, xy,
a, 1
6, 1
= 0,
which may be in like manner considered as the rational equation between x, y.
The preceding results are of course selfevident, but by applying the same process
to the equations
ic + y + z = 0, a^ = ay y*^b, ^ = c,
109] ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. 41
we have results of some elegance. Multiply the equation first by 1, yz, zx, xy, reduce
and eliminate the quantities x, y, z, xyz, we have the rational equation
111 =0;
1 . c 6
I c . a
1 6 a .
and again, multiply the equation by x, y, z, xyz, reduce and eliminate the quantities
1, yz, zx, xy, the result is
I a 6 c = 0,
a . 1 1
6 1.1
ell.
which is of course equivalent to the preceding one (the two determinants are in fact
identical in value), but the form is essentially different. The former of the two forms
is that given in my paper "On a theorem in the Geometry of Position" (Journal,
voL II. [1841] p. 270 [1]): it was only very recently that I perceived that a similar
process led to the latter of the two forma
Similarly, if we have the equations
X'\y + z + w = 0, 5^ = a, ^" = 6, z^=^c, iv'^dy
then multiplying by 1, yz, zXy xy, xw, yw, zw, an/zw, reducing and eliminating the
quantities in the outside row.
X. Vf '.
tr, yzWf zwx, y>xy^ xyz
we have the result
1 1 1
1
• • •
•
1
1
1
c 6
c . a
h a
•
•
•
1 . .
. 1 .
. . 1
d . .
. d .
. . d
a
b
c
. 1 1
1 . 1
1 1 .
•
•
•
• • •
•
a b c
d
= 0;
so if we multiply the equations by x, y, z, w, yzw, zwx, wxy, and xyz, reduce and
eliminate the quantities in the outside row,
C. II. 6
42
ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS.
[109
1, yz, zXt xy^ xto, yw, zwt xytw
we have the result
a
h
c
. 1 1
1 . 1
1 1 .
1 . .
. 1 .
. . 1
•
•
•
d
• • •
1 1 1
c b
c . a
b a
•>
1
1
1
1
9
•
•
d
. d .
. . d
•
a b c
• • •
= 0,
which however is not essentially distinct from the form before obtained, but may be
derived from it by an interchange of lines and columns.
And in general for any even number of quadratic radicals the two forms are not
essentially distinct, but may be derived from each other by interchanging lines and
columns, while for an odd number of quadratic radicals the two forms cannot be so
derived from each other, but are essentially distinct.
I was indebted to Mr Sylvester for the remark that the above process applies to
radicals of a higher order than the second. To take the simplest case, suppose
^ + y = 0, a^=a, y^=^b;
and multiply first by 1, a^y, xy^\ this gives
a? f y . =0
bx
+ aj^y • = ;
or, eliminating.
1 1
a
1
1
0;
next multiply by x, y, a^\ this gives
x"
or, eliminating.
. \xy =
y« + iry =
6a^ + ay* . =0;
1 . 1 =0;
. 1 1
b a
and lastly, multiply by aj*, y\ xy\ this gives
b . + ary> =
. ic*y + ay = ;
109] ON THE RATIONAUSATION OF CERTAIN ALQEBRAICAL EQUATIONS. 43
or, eUminating,
a 1 . =0;
6 . 1
. 1 1
where it is to be remarked that the second and third forms are not essentially distinct,
since the one may be derived from the other by the interchange of lines and columns.
Applying the preceding process to the sjrstem
multiply first by 1, asyz^ c^}^s^^ ^z, y'j?, ah/^ ah/, y*^, s^x, reduce and eliminate the
quantities in the outside row,
«. Vt «» y'«'. ^V't y***» '^^y* ***'» *V
the result is ~^_ I I =0;
1 1 1
• • •
• • •
• • •
111
• • •
• • •
• • •
a b c
. a
b . .
. c .
1 . .
. 1 .
. . 1
. 1 .
. . 1
1 . .
. a .
. . 6
c .
1 . .
. 1 .
. . 1
. . 1
1 . .
. 1 .
next multiply by a?, y, z, yV, s^a^^ a^y\ ahfz^ y*^a?, s^xy, reduce and eliminate the
quantities in the outside row,
the result is ""; I \ =0;
1 . .
. 1 .
. . 1
. 1 1
1 . 1
1 1 .
• • •
• • •
• • •
c 6
c . a
6 a
• • •
• • •
• • •
1 . .
. 1 .
. . 1
• • •
• • •
• • •
a .
. 6 .
. c
. 1 1
1 . 1
1 1 .
6—2
44 ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. [109
lastly, multiply by a^, y*, z\ yz, zx, xy, xt/^s^^ ys^^> xyV^ reduce and eliminate the
quantities in the outside row,
1 xyz, zhjU\ yz\ «x», ary*, y*«, f'x, a?y
the result is I" I = ;
a
b
c
•
•
•
. 1 .
. . 1
1 . .
. . 1
1 . .
. 1
•
1
1
1
1 . .
. 1 .
. . 1
1 . .
. 1 .
. . 1
•
•
•
1
1
1
c
a .
. b .
. 6 .
c
a
where, as in the case of two cubic radicals, two forms, viz. the first and third forms
of the rational equation, are not essentially distinct, but may be derived from each
other by interchanging lines and columns.
And in general, whatever be the number of cubic radicals, two of the three forms
are not essentially distinct, but may be derived from each other by interchanging lines
and columns.
110]
45
110.
NOTE ON THE TRANSFORMATION OF A TRIGONOMETRICAL
EXPRESSION.
[From the Cambridge and Dublin Mathematical Journal, vol. ix (1854), pp. 61 — 62.]
The differential equation
dx dy dz
v = 0,
(o + «) \/(c + a") (a^y)^{c + y) (a + z)>^(c + z)
integrated so as to be satisfied when the variables are simultaneously infinite, gives
by direct integration
and, by Abel's theorem,
1, X, (a H a?) V(c + a?) =0.
1, y, (a + y)V(c + y)
, 1, z, (a + <2r) V(c + ^) !
To show d posteriori the equivalence of these two equations, I represent the deter
minant by the symbol Q, and expressing it in the form
n = lf a + x, (a + a:) V(c + a?)
write for the moment f =. /f j &c. ; this gives
46 NOTE OK THE TRANSFORMATION OF A TRIGONOMETRICAL EXPRESSION. [llO
n =
1. (ac)(l + i). (oc)»(^ + i)
(oc)' f, f» + f p + 1
(oc)* P. f f + 1
fVC»
1. f. f
f^?
1. e f
}
^ (ac )»(g+iy + gfi?g)
fvr
1. f f
or, replacing (, i;, (f by their values, we have identically
1, X, (o + a;) \/(c + a?)
1, y, (p.+y)'J{c+y)
1, z, (o + «)V(c+z)
(c+a;)*(c+y)'(c+g)* f Ajc, / «— c la—c ja—c ja—^ /"~^l
(a  c)* IV €+«"*■ V c+y V c+z V c+«V c+yV c+rj
— c a— c
c+a?' C+J?
— c a— c
/a— c
c+y' c+y
— c a— c
C+8:' C+r
and the equation
/ g — c / g — c / g — c / g — c / g — c /g — c ^ ^
V c + a? V c + y V c + jgr Vc + a;V cTy V cT^
is of course equivalent to the trigonometrical equation
tan a/jHZ + tan' a/^ + tan a/^'' = 0.
V c + « V c + y V c + £^
which shows the equivalence of the two equations in question.
Ill]
47
111.
ON A THEOREM OF K LEJEUNEDIRICHLET'S.
[From the Cambridge and Dublin Mathematical Journal, vol. ix. (1854), pp. 163 — 165.]
The following formula,
is given in LejeuneDirichlet's wellknown memoir "Recherches sur diverses applica
tions &c." (Crelle, t. xxi. [1840] p. 8). The notation is as follows: — On the lefthand
side (a, 6, c), (a', b\ c'), ... are a system of properly primitive forms to the negative
determinant D (Le. a system of positive forms); x, y are positive or negative integers
including zero, such that in the sum Sj*^"*'***^'^*''", aa^ + 2hxy 4 cy* is prime to 2jD,
and similarly in the other sums ; q is indeterminate and the summations extend to
the values first mentioned, of x and y. On the righthand side we have to consider
the form of jD, viz. we have D — PS' or else jD = 2PiSf, where S* is the greatest
square factor in D and where P is odd: this obviously defines P, and the values
of S, €, which are always ± 1 (or, as I prefer to express it, are always ±) are given
as follows, viz.
/) = PS*, P=l(mod4), S, €= + +,
/> = PiS*, P = 3(mod4), S, € = +,
D = 2PiS*, P=l(mod4), S, € = + ,
i> = 2P/S*, P = 3(mod4), S, € = ,
», n' are any positive numbers prime to 2Z), f pj is Legendre's symbol as generalized
by Jacobi, viz. in general if /> be a positive or negative prime not a factor of n,
48 ON A THEOREM OF M. LEJEUNE DIRICHLET's. [Hl
then (  j = + or — according as n is or is not a quadratic residue of p (or, what is
the same thing, p being positive, f— j = w*<**"« (modp)), and for P=pp'p*' .,.,
and the summation extends to all the values of n, nf of the form above mentioned.
In the particular case jD = — 1, it is necessary that the second side should be doubled.
The method of reducing the equation is indicated in the memoir. The following are
a few particular cases.
i) =  1, X^"^^ = 42 ()*<»»« q^^\
2) = — 2, V j«»+«y« = 22 ( — )* <**" +* <****> g"*» ,
or (l + 25« + 25r«425r"...)(g + g* + g"4g^ + ...)
 9 . _?•_ 2! £_ + &c
an example given in the memoir.
i) = 3, 2?*"+'«^=22g)3**«',
+ 2(5'* + ^''^ J^ 4 J»*^4 ...)(3* + g"' + ?~ 4 ?>~ ...)
^g + g* g' + g* , g^ + g" g"*g " ,
lV* 1g" lg« l^"^*"
I am not aware that the above theorem is quoted or referred to in any sub
sequent memoir on Elliptic Functions, or on the class of series to which it relates;
and the theorem is so distinct in its origin and form from all other theorems relating
to the same class of series, and, independently of the researches in which it originates,
so remarkable as a result, that I have thought it desirable to give a detached state
ment of it in this paper.
112]
49
112.
DEMONSTRATION OF A THEOREM RELATING TO THE
PRODUCTS OF SUMS OF SQUARES.
[From the Philosophical Magazine, vol. iv. (1852), pp. 515 — 519.]
Mb Ktrkman, in his paper " On Fluquatemions and Homoid Products of Sums of
n Squares" {Phil, Mag. vol. xxxiii. [1848] pp. 447 — 459 and 494 — 509), quotes from a
note of mine the following passage: — "The complete test of the possibility of the pro
duct of 2* squares by 2** squares reducing itself to a sum of 2^ squares is the following :
forming the complete sjrstems of triplets for (2^ — 1) things, if eab, ecd, fac, fdh be any
four of them, we must have, paying attention to the signs alone,
(±eab)(±ecd) = (±f(w)(± fdh) ;
Le. if the first two are of the same sign, the last two must be so also, and vice versd;
I believe that, for a system of seven, two conditions of this kind being satisfied would
imply the satisfsu^tion of all the others : it remains to be shown that the complete system
of conditions cannot be satisfied for fifteen thinga" I propose to explain the meaning
of the theorem, and to establish the truth of it, without in any way assuming the exist
ence of imaginary units.
The identity to be established is
(ti;* + o« h 6» + ...) (w; + a; + b; ...) = w,; + a,; + b,; + ...
where the 2* quantities w, a, 6, c, ... and the 2** quantities w,, a„ 6„ c,, ... are given quan
tities in terms of which the 2^ quantities w,^, a,,, 5,,, c,„ ... have to be determined.
Without attaching any meaning whatever to the symbols a^, 6^, c^ ... I write down
the expressions
w + aa^ + bb^ + cc^..., w, + afl^ + bf>^ + c,c^ . . . ,
an. 7
50 DEMONSTRATION OF A THEOREM RELATING [112
and I multiply as if a^, 6^, c^... really existed, taking care to multiply without making
any transposition in the order inter se of two symbols a^, 6^ combined in the way of mul
tiplication. This gives a quasiproduct
tvw, + (aw, + a,w) a^ + (btv, + b,w) 6^ + • • •
+ aafl^* + bb,b^* + . . .
{ab,aX + (^fibo<^o + ""
Suppose, now, that a quasiequation, such as
means that in the expression of the quasiproduct
be, c a % d b , b , a c , b d
are to be replaced by
fl, 0, c, — ci,— 6, — c I
*^o' o' o* o* o' o *
and that a quasiequation, such as ajb^c^ = — , means that in the expression of the quasi
product
6c. cd. a b , c b , ac. b (i
are to be replaced by
^o» ""^o> "c^» ^o> Ky Co
It is in the first place clear that the quasiequation, ajbjc^ = +, may be written in
any one of the six forms
a6c=4, 6c(i=4", ca6=4,
a c b =^ —, c6a=5:— , ((20= — ;
^0^0 o » *^o*'o o ' o**© o »
and so for the quasiequation ajb^c^ =» — . This being premised, if we form a system of
(}uasiequations, such as
<^oMo = ±» ao^o«o = ±» &c
where the system of triplets contains each duad once, and once only, and the arbitrary
signs are chosen at pleasure; if, moreover, in the expression of the quasiproduct we
replace a,', 6^', ... each by — 1, it is clear that the quasiproduct will assume the form
^/y» ^y/» K* C// ••• b^iiig determinate functions of w, a, b, c, ...; w,, a,, b„ c, ..., homo
geneous of the first order in the quantities of each set ; the value of w,, being obviously
in every case
w,^ = vm^ — tta, — 66, — cc, . . . ,
and a,,, b„, c,,,... containing in every case the terms aw,{a,w, btv, + b,w, cw,¥ cjw,.,, but
the form of the remaining terms depending as well on the triplets entering into the
112] TO THB PRODUCTS OF SUMS OF SQUARES. 51
system of quasiequations as on the values given to the signs ± ; the qtiosiegtuxtions
serving, in fact^ to preset^ a rule for the formation of certain functions w,„ a^„ 6,,, c,,, ...,
the properties of which functions may afterwards be investigated.
Suppose, now, that the system of quasiequations is such that
e a b , e c d
O O O' "O^O O
being any two of its triplets, with a common symbol e^, there occur also in the system
the triplets
f fi c , f d b , Odd, a b c \
and suppose that the corresponding portion of the system is
^o^o^o = €, e^c^d^ = c',
/o«oCo = (r, fodoK = ^»
where €, f, *, e', ^, t each of them denote one of the signs + or — ; then e,„f„, g,, will
contain respectively the terms
€ (ab,  a,b) + e' (cd,  c^d),
i (ad^ — a,d) + if (be, — bfi) ;
and e,,^+f*+g,* contains the terms
(a> 4 6» + c» + d«) (a; + b; + c/ + d;)  aJ'a;  H;  cFc^dl'd;
+ 2 [ee {ab,  ajb) (cd,  c,d)
+ ^(ac,afi){db,'dfi)
+ u' {ad,  a,d) (be,  bfi)] ;
and by taking account of the terms ew, + e,w, fw,+f,w, gw,\g,w in e„, f,„ gr^/ respect
ively, we should have had besides in ^^ ^f^ + g^ ^® terms
\2(ee,+ff^ + gg,)ww,.
Also k;^' contains the terms
ii/Hv,*\'a*a,^ + b'b; + (^; + dM,^
i(ee,+ff,+gg,)ww,]
whence it is easy to see that
(ti;* + a« + 6« + c*+ ...)(w/ + a/ + V + c/ + ...)
+ 2S [ee' (a&,  ajb) (cd,  c^d)
+ ^(ac,a,c)(db,d,b)
+ ii' (ad, — a,d) (be, — 6^c)].
7—2
52 DEMONSTRATION OF A THEOREM RELATING &€. [112
where the summation extends to all the quadruplets formed each by the combination
of two duads such as ab and cd, or ac and db, or ad and be, L e. two duads, which, com
bined with the same common letter (in the instances just mentioned e, or /, or g), enter
as triplets into the sjrstem of quasiequations — so that if i/ = 2** — 1, the number of quad
ruplets is
i{H''l)i(''3)}v.i, = Av(.; !)(,; 3).
and the terms under the sign 2 will vanish identically if only
but the relation e^ = a' is of the same form as the equation cc' = ff ; hence if all the
relations
are satisfied, the terms under the sign S vanish, and we have
{w^; 4 a^; + 6,/ + c,;+ ...) = («^ + a' + 6" + c«+ ...) (w; + a; + 6/ + c/ + ...)
which is thus shown to be true, upon the suppositions —
1. That the sjrstem of quasiequations is such that
6 db ^ 6 c d
being any two of its triplets with a common sjrmbol e^, there occur also in the sjrstem
the triplets
2. That for any two pairs of triplets, such as
e a b , 6 c d and f a c , f d b ,
the product of the signs of the triplets of the first pair is equal to the product of the
signs of the triplets of the second pair.
In the case of fifteen things a, 6, o, ... the triplets may, as appears from Mr Kirk
man's paper, be chosen so as to satisfy the first condition; but the second condition
involves, as Mr Kirkman has shown, a contradiction; and therefore the product of two
sums, each of them of sixteen squares, is not a sum of sixteen squares. It is proper to
remark, that this demonstration, although I think rendered clearer by the introduction of
the idea of the system of triplets furnishing the rule for the formation of the expres
sions w,,, a,,, 6,„ c^,, &a, is not in principle different fiom that contained in Prof. Young's
paper "On an Extension of a Theorem of Euler, &c.", Irish Transactions, vol. xxi. [1848
pp. 311—341].
113]
53
113.
NOTE ON THE GEOMETRICAL REPRESENTATION OF THE
INTEGRAL jdx^ >/(x + a) ~(aj + h){x + c).
[From the Philosophical Magazine, voL v. (1863), pp. 281 — 284.]
The equation of a conic passing through the points of intersection of the conies
a^ + y* + ^ = 0,
is of the form
w (ic* + y* + 2:*) + cut" + 6y« + ag« = 0,
where k; is an arbitrary parameter. Suppose that the conic touches a given line, we
have for the determination oi w b, quadratic equation, the roots of which may be
considered as parameters for determining the line in question. Let one of the values
oi w he considered as equal to a constant quantity A;, the line is always a tangent
to the conic
*(«" + y* + ^) + flwj* + 6y' + C4^ = ;
and taking w^p for the other value of t^;, /) is a parameter determining the parti
cular tangent, or, what is the same thing, determining the point of contact of this
tangent
The equation of the tangent is easily seen to be
X ^hc '/a + k VoTp + y'</c — a^b + k '^b+p^z 'Jah "Jc + k '^cTp^ ;
suppose that the tangent meets the conic a^ + y^+z^ = (which is of course the
conic corresponding to w = oo ) in the points P, P', and let d, oo be the parameters
of the point P, and ^, oo the parameters of the point P', i.e. (repeating the defini
54 NOTE ON THE GEOMETRICAL REPRESENTATION OF [ll3
tion of the terms) let the tangent at P of the conic a^Hy' + £^ = be also touched
by the conic d(a^ + y' + ^') + aa?» + 6y» + 0^ = 0, and similarly for ff. The coordinates
of the point P are given by the equations
x:y:z=^ 'Jbc Va + d: ^ca 'JbTO : Va6 Vc + tf ;
and substituting these values in the equation of the line PP", we have
(b  c) Va + A VoTp Va + d + (00) ^bTk y/b+p ^/W^ + (a  6) \/c^k y/c+p Vcf (?
= 0. ..(♦),
an equation connecting the quantities p, 0. To rationalize this equation, write
V(a + i) (a +p)~(a + d)~= X + /ui,
V(6 4A)(6+i>)(6 + d) = X + ^,
values which evidently satisfy the equation in question. Squaring these equations, we
have equations from which X', \p^ fj} may be linearly determined ; and making the
necessary reductions, we find
X'* = a6c + kpd,
 2X/i= 6c + ca + a6(p^4Ap4 kd\
or, eliminating X, /[t,
{6c + ca + a6(p5 + ip4A;d)}'4(a46 + c + A?+p + d)(a6c + A:/>d) = 0, (♦),
which is the rational form of the former equation marked (*). It is clear from the
symmetry of the formula, that the same equation would have been obtained by the
elimination of Z, M from the equations
V (jfc + a) (A + 6) (Jfc + c) = Z + Mk,
V(p + a)(/> + 6)(jp + c) ^L + Mp,
and it follows from Abel's theorem (but the result may be verified by means of
Euler's fundamental integral in the theory of elliptic functions), that if
dx
J • V(a! + o) (a + 6)(a! + c) '
then the algebraical equations (*) are equivalent to the transcendental equation
±m±nj)±nd=0;
113] THE i^n:m}RAh jdx^J{x + a){x + b){x + c). 55
the arbitrary constant which should have formed the second side of the equation
having been determined by observing that the algebraical equation gives for p^d,
& = 00 , a system of values, which, when the signs are properly chosen, satisfy the
transcendental equation. In fact, arranging the rational algebraical equation according
to the powers of i, it becomes
*»(p  ey  2k {p^(p + ^) + 2 (a + 6 4c)pe + (fcc + ca 4 ah){p + ^) + 2abc]
\'jfe'''2{bc{ca'\ah)pd'' 4a6c (jp + d) + fr^c* + c»a» + a^6'  2a^bc  26«ca  2c^ab = ; (♦)
which proves the property in question, and is besides a very convenient form of the
algebraical integral. The ambiguous signs in the transcendental integral are not of
course arbitrary (indeed it has just been assumed that for p = 0y Up and RO are to
be taken with opposite signs), but the discussion of the proper values to be given
to the ambiguous signs would be at all events tedious, and must be passed over for
the present.
It is proper to remark, that d=p gives not only, as above supposed, &=x,
but another value of k, which, however, corresponds to the transcendental equation
±Uk± 2np = ;
the value in question is obviously
, _ fi*  2 (6c+ eg + ab)p^  Sahcp + 6V + Ca' + a'6^ ~ 2a»6c  2fe'co 2(^06
"" (p + a) (p + 6) () + c)
Consider, in general, a cubic function aai^ + Sba^y + 3ca:y* 4 dy*, or, as I now write
it in the theory of invariants, (a, b, c, d) {x, yY, the Hessian of this function is
(acb», Hadbc), bdr.c>)(a?, y)>,
and applying this formula to the function (p 4 a) (p 4 6) (p 4 c), it is easy to write
the equation last preceding in the form
lA' D (a\b\c^ ^ Se8aiB,n {(p + a)(p 4 b)(p 4 c)}
^^~^ (a464c) (p4a)(p46)(p + c)
which is a formula for the duplication of the transcendent Ilx,
Reverting now to the general transcendental equation
±nA:±IIp±nd = 0,
we have in like manner
±ni±np±n^=0;
and assuming a proper correspondence of the signs, the elimination of Tip gives
56 NOTE ON THE GEOMETRICAL REPRESENTATION &C. [ll3
i.e. if the points P, P* upon the conic a;* + y' + 2r* = are such that their parameters
6, ff satisfy this equation, the line PP will be constantly a tangent to the conic
k {x^ + y' 4 z") + {aa? + 6y« + c^) = 0.
Hence also, if the paraibeters ^^ k\ Id' of the conies
k (aJ» + y' + 2») + cw^ + 6y* + c^ = 0,
k' (a^ + y' + ^*) + cur» + 6y»+C4:» = 0,
A"(«* + y» + ^«) + (m;» + 6y> + c^» = 0,
satisfy the equation
nA: + nA' + nr = o,
there are an infinity of triangles inscribed in the conic a;* + y' + j* = 0, and the sides
of which touch the lastmentioned three conies respectively.
Suppose 2nA; = II/c (an equation the algebraic form of which has already been
discussed), then
= CO gives ff ^ K\ or, observing that ^ = oo corresponds to a point of intersection
of the conies a^ + y" + £^ = 0, cw^ 4 6y' + C2^ = 0, k is the parameter of the point in
which a tangent to the conic A:(a;' + y* + »®) + a^ + 6y' + c^ = at any one of its
intersections ¥dth the conic aJ'hy* + 4:* = meets the lastmentioned conic. Moreover,
the algebraical relation between 0, ff and k (where, as before remarked, /v is a given
function of k") is given by a preceding formula, and is simpler than that between
d, ff and k.
The preceding investigations were, it is hardly necessary to remark, suggested by
a wellknown memoir of the late illustrious Jacobi, and contain, I think, the extension
which he remarks it would be interesting to make of the principles in such memoir
to a system of two conies. I propose reverting to the subject in a memoir to be
entitled "Researches on the Porism of the in and circumscribed triangle." [This was, I
think, never written.]
114]
57
114.
ANALYTICAL RESEARCHES CONNECTED WITH STEINER'S
EXTENSION OF MALFATTI'S PROBLEM.
[From the Philosophical Transaciiona of the Royal Society of London, vol. CXLII. for the
year 1852, pp. 253—278: Received April 12,— Read May 27, 1852]
The problem, in a triangle to describe three circles each of them touching the two
others and also two sides of the triangle, has been termed after the Italian geometer
by whom it was proposed and solved, Malfatti's problem. The problem which I
refer to as Steiner^s extension of Malfatti's problem is as follows: — "To determine
three sections of a surface of the second order, each of them touching the two others,
and also two of three given sections of the surface of the second order," a problem
proposed in Steiner's memoir, "Einige geometrische Betrachtungen," CreUCy t. i. [1826
pp. 161 — 184]. The geometrical construction of the problem in question is readily
deduced from that given in the memoir just mentioned for a somewhat less general
problem, viz. that in which the surface of the second order is replaced by a sphere ;
it is for the sake of the analytical developments to which the problem gives rise, that
I propose to resume here the discussion of the problem. The following is an analysis of
the present memoir: —
§ 1. Contains a lemma which appears to me to constitute the foundation of the
aaalytical theory of the sections of a surfietce of the second order.
§ 2. Contains a statement of the geometrical construction of Steiner's extension
of Mal£Eitti's problem.
§ 3. Is a verification, founded on a particular choice of coordinates, of the con
struction in question.
§ 4. In this section, referring the surface of the second order to absolutely general
coordinates, and after an incidental solution of the problem to determine a section
touching three given sections, I obtain the equations for the solution of Steiner's
^tension of Malfisitti's problem.
an. 8
58 ANALYTICAL RESEARCHES CONNECTED WITH [ll4
§ 5. Contains a separate discussion of a sjrstem of equations, including a^ a
particular case the equations obtained in the preceding section.
^ 6 and 7. Contain the application of the formulae for the general system to the
eijuations in § 4, and the development and completion of the solution.
§ 8. Is an extension of some preceding formulae to quadratic functions of any
number of variables.
§ 1. Lenwia relating to the sections of a surface of the second order.
If
flWJ" + fry* + c^ + dtu>' + 2fyz h 2gzx + 2hxy + 2lxw h 2mt/w h 2nzw =
be the equation of a surface of the second order, and
the reciprocal equation, the condition that the two sections
\'x h fiy h v'z h p^w = 0,
may touch, is
(ax* h iSfi^ + ©!/» + Bp' h 2JpMi' + 2(Sv\ + i'f^XfjL + 2%\p h i^fip h 2^vp)^
X (av» + i8^'» + €v' + B/» + 2jffiV + 2(SvX + 2f^\v + 2'a\y h 2iW/tv + 2^v'py
+ U(Xp' h \'p)hMW +M» + ia (vp'^v'p)):
and in particular if the equation of the surface be
cux^ \ bt/* \ cz^ \ 2fyz h 2gzx h 2hay\pvj^ = 0,
the condition of contact is
ax« +i8/i» +ffii/» +2JP/IAI/ h2CBi/X+2^X^hp»j
(ax'« +i8M« +€^'' +2JF /iV + 2flri.V + 2?^XV + ^p
in which last formula
a=6c/', 13*oa>5r», aD=o6A»,
§ = ghaf , (S = 1ifbg. ^=fgch,
K = ahc  of*  bf  ch* + 2/gh.
114] steimer's extension of malfatti's pboblem. 59
§2.
In order to state in the most simple form the geometrical construction for the
solution of Steiner's extension of Malfatti's problem, let the given sections be called
for conciseness the determinators ^ ; any two of these sections lie in two different cones,
the vertices of which determine with the line of intersection of the planes of the
determinators, two planes which may be termed bisectors ; the six bisectors pass three
and three through four straight lines ; and it will be convenient to use the term
bisectors to denote, not the entire system, but any three bisectors passing through the
same line. Consider three sections, which may be termed tactors, each of them touching
a determinator and two bisectors, and three other sections (which may be termed
separators) each of them passing through the point of contact of a determinator and
tactor and touching the other two tactors ; the separators will intersect in a line which
passes through the point of intersection of the determinators. The three required
flections, or as I shall term them the resultors, are determined by the conditions that
each resultor touches two determinators and two separators, the possibility of. the
ooDstruction being implied as a theorem. The d posteriori verification may be obtained
as follows: —
§ 3.
Let J? = 0, y = 0, « = be the equations of the resultors, w; = the equation of the
polar of the point of intersection of the resultors. Since the resultors touch two and
two, the equation of the surfisice is easily seen to be of the form
2yz + 2zj? h 2xy + k/« = 0. («)
The determinators are sections each of them touching two resultors, but otherwise
arbitrary; their equations are
ax + ^^y + ^z^w^O,
The separators are sections each of them touching two resultors at their point of
contact (or what is the same thing, passing through the line of intersection of two
resultors), and all of them having a line in common. Their equations may be taken
to be
cy — 62: = 0, cw^ — ca? = 0, 6a?  ay = 0,
M use the words ** detenninatorB,'' <ftc. to denote indifferently the sections or the planes of the sections;
the context is always saffident to prevent ambignity.
* The reeiprooal form is, it should be noted,
8—2
60 ANALYTICAL RESEABCHES CONNECTED WITH [114
the values of a, 6, c remaining to be determined. Now before fixing the values of
these quantities, we may find three sections each of them touching a detenninator at
a point of intersection with the section which corresponds to it of the sections
cy — 6? = 0, aZ'ca) = 0, for — ay = 0, and touching the other two of the lastmentioned
sections ; and when a, 6, c have their proper values the sections so found are the
tactors. For, let \x\fMy\vz + f>w = be the equation of a section touching the deter
minator — airh^yh^^ + t£; = 0, and the two sections &c — ay = 0, az — cx = 0: and
suppose
A« = X« + m' + »^  2/Av  2i/X  2X/I  2p» ;
the conditions of contact with the sections 6a? — ay = 0, ae — ca? = are found to be
(6 4 a) A = (6 + a) \  (6 + a) ft  (6  a) I/,
(c + a) A = (c + a) X — (c — a) fi  (c + a) I/,
values, however, which suppose a correspondence in the signs of the radicals. Thence
(b \ a) fi = (c + a)v ; or since the ratios only of the quantities \ fi, v, p are material,
fi = c '\' a, j^ = 6 + a, and therefore
A« = X«  2 (2a h 6 4 c) X + (6  cy  2p«, = (X  6  cf,
or p» = — 2 (aX + 6c).
Hence the equation to a section touching 6a? — ay = 0, a^ — ca? = is
Xj?4(c + a)y + (6 + a)irhV 2(aX + 6c)\ w = ;
and to express that this touches the detenninator in question, we have
± a (X  6  c) = (a + ) X  a (2a + 6 h c) + 2 V2(aX + 6c) ;
and selecting the upper sign,
1 J
 X2aa = 2V2(aX + 6c);
whence
X =  2a {aa^ " 26c), V  2 (aX h 6c) = (2aa  V ^6c) ;
or the section touching the determinator and the sections 6a? — ay = 0, a? — ca? = is
2a(aa V26c)«rh(c + a)y + (6ha)? + (2aa>/26c)w = 0;
and at the point of contact with the determinator
2y« + &?« + 2ay + «;• = 0.
114] steiner's extension of malfatti's problem. 61
Eliminating w between the first and second equations and between the second and
third equations,
V26c (ax h  y 4 — ^^ + cy + 6? = 0,
and from these equations (cy — bzy = 0, or the point of contact lies in the section
cy — bz^O. It follows that the equations of the tactors are
2a(aaV26c)a? + (c + a)y + (6 + a)8:+(2aa V'  26c) w = 0,
(c + 6) a:  2/8 (6/8  V  2ca) y + (a + 6) ? + (26/8  V  2ca) w = 0,
(6 + c) a: + (a + c) y  27 (07  V 2a6) ? h (2c7  V2a6) w = 0,
where a, b, c still remain to be determined.
Now the separators pass through the point of intersection of the determinators ;
the equations of these give for the point in question,
X : y : z : «; = (2/87 h 1) ( a + /8 + 7 + 2a^7)
(27a +1)( a/8 + 7 + 2a/87)
(2a/3 + l)( a4/37 + 2a/37)
4a»/8Vl+a» + /9» + 7«;
and the values of a, 6, c are therefore
a :b : c = (2/87+ 1) (a + i8 + 7 + 2a/37)
:(27a+l)( a  /8 + 7 + 20/87)
:(2a/8 + l)( a + /87 + 2a/87),
which are to be substituted for a, 6, c in the equations of the separators and tactors
respectively.
Now proceeding to find the bisectors, let Tix + fiy + vz + pw^O be the equation
of a section touching the determinators,
^«/3y + ^* + w0. ^x + ^yyi + w = 0;
and suppose, as before, A**X*hfi* + i^ — 2/ij^ — 2i/X — 2X/i — 2p*; the conditions of con
tact are
±/8A = /3X(/8 + i)/i + /8i/2p,
T 7A = 7X + 7/1 — f 7 +  j V — 2p,
62 ANALYTICAL RESEARCHES CONNECTED WITH [114
where it is necessary, for the present purpose, to give opposite signs to the radicals.
For if the radicals had the same sign, it would follow that
^^i8\(/3 + ^)Mh/3i/2pJ^[^7Xh7/i(7H^),;2pj=0;
hence the section \x + fiy { vz \ pw =^ would pass through the point
nil 22
or the section would be a tangent section of the two determinators of the same
class with the resultor a? = 0, which ought not to be the case. The proper formula is
^^/3X(^ + ~)AiH/3i/2p]+^[^7\ + 7/i(7 + ^)i'2p]=0;
and this equation being satisfied, the section
passes through a point
«o 112 2
a: : y . z : w  z : gi^y'^"*
The bisector passes through this point and the line of intersection of the determi
nators ; its equation is
l^xfiy + lz + wy^{^x + ^yyz + w)=0;
or reducing and completing the system, the equations of the bisectors are
In order to verify the geometrical construction, it only remains to show tliat
each bisector touches two tactors. Consider the bisector and tactor
(^ + 2^)«+(l + 2^)y + (2^2')' + («)«' = ^' ■
 2a (aa  V26c) ar + (c + a)yh(6ha)^ + (2aa V2ic) w = 0;
and represent these for a moment by
Xa? h fiy + 1/? + /Mi; = 0, X'a? + //y + p'z + p'w = ;
114] steiner's extension of malfatti's problem. 63
if A be the same as before, and A' the like function of \\ ja\ v\ p, also if
* = XX' + /i/A' h w'  (/lAi/' + fi'v)  (vV + v\)  (X/ + X»  2pp',
then
^■(^^s)"'
A'* = (200?  2a V  26c + 6 + c)»,
and the condition of contact AA' = <I> {taking the proper rign for the radicals) be
comes
or reducing,
"*^^ + ''2^1 = ^'
an equation which is evidently not altered by the interchange of a, a and 6, fi. The
conditions, in order that each bisector may touch two tactors, reduce themselves to
the three equations,
*'+*2^i'^'^=<*'
which are satisfied by the values dbove found £nr tk^ quantities a, h, c. The possi
hility and truth of the geometrical construction are thus demonstrated.
Let it be in the first instance proposed to find the equation of a section touch in^^
all or any of the sections a? = 0, 3r = 0, £ = of the surfi^^ of the second order,
^wc" + 6y" + ci^ + Sfyz + 2gzx h thxy + pw?'^ = 0.
Any section whatever of this surface may be written in the form
(aX + Afi + gv) x + (h\ + i/i h /i/) y + (g\ {ffi + ci/) ^ + V — ;> V w = 0,
where
V« =:aX«+ V Hcr' + 2//iav + igvX f 2A\/i  K,
64 ANALYTICAL RESEARCHES CONNECTED WITH [ll4
and \ fi, V are indeterminate. And considering any other section represented by a
like equation,
{pX + hfi'\gi/) X h (AX' h bfi' +>') y + (flfV +// + cv') z + V^ V'w = 0,
where
V'« = aV» + 6/» + ci;'» + 2//j/' + 2fln^ V + 2A\y  K,
it may be shown by means of the lemma previously given, that the condition of
contact is
aXX' + 6/a/a' + cvv' \f(p,v' + iLv) + g (i/X' + v\) + A (X/ia' + X» ± iT = VV.
Suppose that X', y!^ v' satisfy the equations
AV + 6/+yj/' = 0,
gfX' +// + CI/' = 0,
so that the lastmentioned section becomes a; = 0; and observing that the first of
the above equations may be transformed into
/» 1^ CBf
it is easy to obtain X' = v W, /*' = "Tg , v' = j^ . The condition of contact thus becomes
K
and taking the under sign, X = a/^I, so that if in the above written equation we
establish all or any of the equations X = V^, fi = V3B, v = V®, we have the equation
of a section touching all or the corresponding sections of the sections
«; = 0, y = 0, z = 0.
In particular we have for a solution of the problem of tactions, the following
equation of the section touching a? = 0, y = 0, jf = 0, viz.
+ — .
Anticipating the use of a notation the value of which will subsequently appear,
or putting
f=^l^/vgB®JF, g=ys7vH3ffi; h = ^Vvi3B^, j=V24'gti3ar,
114] steiner's extension of malfatti's problem. 65
values which give
ir« = f*g*h* + 2g»h« + 2h«f« + 2f»g«^^',
the equation of the section in question is
7^<f^ + g'hh')a:h;^(Pg'hh')yh^(Phg'~h')^h 'g^''/'''' i(; = 0.
I proceed to investigate a transformation of the equation for the section with an
indeterminate parameter X, which touches the two sections y = 0, z=^0. We have
aV» = (oX + V + 9^y + (^/^^ + i8i/"  2 jp/ii/)  J8ffi + Jp ;
or putting for ^ and v their values VS, V(!D in the second term,
aV'* = (aX + A/A + 5ri/)» + (VS®  Jp)» ;
and introducing instead of X an indeterminate quantity X, such that
aX + A/i + (7J^ = (VS®jp)Z,
we have
also introducing throughout X instead of X, and completing the substitution of V33, v CD,
for /A, V, the equation of the section touching y = 0, ^r = 0, becomes
{aX'^hy'\'gz)X'ky V® + ^ V:JB + w V op Vl h Z» = 0.
It may be remarked here in passing, that this is a very convenient form for the
demonstration of the theorem; "If two sections of a surface of the second order touch
each other, and are also tangent sections (of the same class) to two fixed sections,
then considering the planes through the axis of the fixed sections and the poles of
the tangent sections, and also the tangent planes through this axis, the anharmonic
ratio of the four planes is independent of the position of the moveable tangent sections;"
where by the axis of the fixed sections is to be understood the line joining their poles.
The sections touching ? = 0, a?=0, and iF = 0, y = 0, are of course
x^kQix + hy \'fz) Y\z Va + w VfcpVlf r« = 0,
a? VS +y Va + (flra; h/y + c?) Z + w "J ^cp Vl +i^''= 0,
where
Ax' +6/i' +/i;' =(V^a<!B)7, X' = Vir / =/. »'' = V®,
The conditions of contact of the sections represented by the above written equations
would be perhaps most simply obtained directly from the lemma, but it is proper to
deduce it firom the formula for contact used in the present memoir. If for shortness
<I>(±) = aX'X" + 6/iV' h cv V +/(/!/" h Ai'V) + g {vX' + i/'V) + A (XV + X'V) ± K.
C. II. 9
66 ANALYTICAL BBSEABCHES CONNECTED WITH [114
where the symbol 4>(±) is used in order to mark the essentially different character
of the results corresponding to the different values of the ambiguous sign, then
bap (_) =/(Ax' + V +>') (3^" +//*" + <»").
+ 0SLv'  «SX' ) (g\" +/fi," + cp").
+ (a/'  I^X") ihX' + 6/ +>' ).
+ v'fi" ifSf)+ v'\"f^ + xy/ffi + XV (K f$)
= /(AX' + bfi,' +Jv') ig\" ^fy!' + cv")
+ Va (Va®  <S) O^X" +//*" + (»")
+ Va ( V^S  IQ) (AX' + hy! + >' )
+/C a ^s®'+ 1^ v®if + ffi VI0B  a jF  («5?^  a jF))
= /(AX' + 6/*' +/;') (jrX" +//*" + CI/")
+ v^ (Va®  ffi) («7X" +//' + CI/")
+ va (Vi®  1^) (Ax' + 6/ + >' )
/(VS®  (5) (Va23  ?^),
that is. 6c<I)() = (Va®<ffi)(VaS^){/FZ + V'a(F+Z)/l.
What, however, is really required", is the value of 4>(+); to find this, we have
6c4> (+) = 6c^ () + 26cir
=(V^(ff(E)(vai8i^)(/Fz+va(F+z)+/}
+ 26cir  2/(Vaff  (K) (VaS  1^),
the second line of which is
2 (Va®  «i!) (v^is  1^) 1^ (vn + ®) (vas + 1^) /
^2 (Va@(SHVal8i^) ^(^^^^^(^^^^)_^^^^^,
= 2 (Va®  <!E)(>/ii9  1^) va^,
1 It may be shown without diffioolty that the () ngn would imply that the sections toaching 2 = 0, x=0,
and £=0, y=0, were sections toaching ;c=0 at the same point. By taking the () sign in each equation we
should have the solution of the problem **to determine three sections of a surface of the second order, the two
sections of each pair touching one of three given sections at the same point," which is not without interest ;
the solution may be completed without any difficulty.
114] steiner's extension of malfatti's problem. 67
where
and consequently
iK5<i>(+)==(V^(aDffi)(VaSf^){/Fz+va(F+z)+/+2tfVa},
a reduction, which on account of its peculiarity, I have thought right to work out in
full.
The condition of contact is
4> (+) = VT'' = JL ( VS(^  ffi) ( vai8  1^) vr+T^ ViTzi
Hence finally, the condition in order that the sections
(the former of which is a section touching £^ = 0, d; = 0, and the latter a section touching
ir = 0, y = 0) may touch, is
/Fz+va(F+z)+(/+2^va)V6^vrTT^vnr^=o.
The preceding researches show that the solution of Steiner's extension of Malfatti's
problem depends on a system of equations, such as the system mentioned at the
commencement of the following section.
§5
Consider the sjmtem of equations
a +i8 (F+z)+7 Yz j^h vnnr»vr+z» =0,
«" + y8" (Z + F) + 7"ZF+ 8" Vr+X» VTTP = ;
these equations may, it will be seen, be solved by quadratics only, when the coefficients
satisfy the relations
/3 yS' /3"
7a ya' V'«"'
,y»_a» " 7'»a^ 7"»a"»
9—2
68 ANALYTICAL RESEARCHES CONNECTED WITH [ll4
It may be remarked that these equations are satisfied by
/8 = 0. /3' = 0, /3" = 0. 7 = 8, 7'=S', 7" = ^
or if we write
a , o' a"
 = — ^ , = — m, 77 = — w,
7 7 7
the equations become by a simple reduction,
7»hZ*+2i FZ = P 1,
Z*h Z«+ 2mZZ = m»  1,
Z»+Ph2nZF=n» 1,
which are equivalent to the equations discussed in my paper " On a system of Equations
connected with Malfatti's Problem and on another Algebraical System," Cambridge and
Dublin Mathematical Joumat, t. iv. [1849] pp. 270 — 275, [79]; the solution might have
been effected by the direct method, which I shall here adopt, of eliminating any one
of the variables between the two equations into which it enters, and combining the
result with the third equation.
Writing the second and third equations under the form
^" + 5"z + cvr+z* = 0,
the result of the elimination may be presented in the form
which is most easily obtained by writing X=tan0 and operating with the symbol
cos^^; but if the rationalized equations be represented by
V + 2/Z + i;'Z» = and V' + 2/'Z + i/''Z» = 0,
the form
4 {W  /i'«) {\%"  /'») = {W + \'V  2//i'7
leads easily to the result in question. The values which enter are
(7 = 8' VIT^, C" = S" VTTF';
whence, in the first place, by the equation connecting F, Z,
C(7" = ?l'{a + ^(F+Z)h8FZ}.
114] steinbr's extension of malpatti's problem. 69
It is obviously convenient that A'A"\BB' should be symmetrical with respect to
F and Z, and this will be the case if
that is, if /8'(y'a'0=/3"(7'«');
or assuming that the equations are symmetrically related to the system, we have the
first set of relations between the coefficients, relations which are satisfied by
a = 7 + 20i8, a' = y + 20i8', a'' = 7" + 20/8",
and the values of a, a\ a" will be considered henceforth as given by these conditions.
We have
il'^'' + 5'^''  C^C^' = o'o'' + /8'/8'' + (7'/3'' + 7^/8' + 20/3'i8'O ( F + Z) + (^^^
H^{a + ^(FhZ)H7FZ}.
The quantities A'^^B^— C\ A"^ + B'^ — (7'» are quadratic functions of Z and Y respectively,
and with proper relations between the coefficients, we may assume
^A'^'^B^'C'^){A'''''^B'''^G'^)^W{V^^kl{a^fi{Y^Z)^r^YZy
in which 17 is a linear function of Y '\ Z and FZ, and h and U are constants. The
first side must, in the first place, be symmetrical with respect to F and Z, or
must be proportional to
But since
(«' + 7')/8'. (a" + 7")/9"
are proportional to
it is only necessary that
should be proportional to
y«.a'«, 7"»a"»,
^a + yiS'*, /3"« + y'«S"»
y«a'», 7"*a"»;
or since the equations are supposed symmetrically related to the system, we must have
the second set of relations between the coefficients. Suppose
then since
ya» = 4(7 + 0/8)0)8, &c..
70 ANALYTICAL RESEARCHES CONNECTED WITH [114
we have 8» «/8« +7» *8(y +^fi)fi
and S, S\ S"' will be supposed henceforth to satisfy these equations.
We have next
which may be simplified by writing
u  <& l\ii4>
g as t ^ w :s 2—
where ^ v are to be considered as given functions of a and <f>. These values give
A'* +R*Cr* =4(7' +4>$')^s {Z + ,i){Z + v),
^"»+ £"«_ C"t = 4 (y + 0y3") ff'8{Y+it) (F+ 1/).
Hence, putting for simplicity
we have
4(Z + /*)(Z+i;)(F+At)(F+v)=£7» + *[(a+/3(F+Z) + 7F^S'(l + F»)(l+Z')];
and the two sides have next to be expressed in terms of T + Z and YZ.
If for symmetry we write
f = l. 7, = 7+Z, K^YZ,
then
and U' is now to be considered a linear function of f, 17, 2^.
The condition that the first side of the equation may divide into factors, gives an
equation for determining k ; since the condition is satisfied for A; = and k<x>, the
equation will be linear, and it is easily seen that the value \a k^^iji — vf. In fact
hence
{2,*vf + 0* + v) 1; + 2f }«  t7» = ^^i^ {(af + /3i, + 7f )•  S* (f + ?)■}.
114] steineb's extension of malfatti's problem. 71
and we may assume
2Mvf + 0* + .;)i7 + 2?O=^^{(af + ^i, + y?) + 8(f + ?)},
subject to its being shown that
gives a constant value for A. The comparison of coefficients gives
*~ a
the first and third of these give
{(A.l),(Ai)4;
4 (1  ^v) = ^g" (a + ^) (7  «),
which will be identical with the second, if
2(1 Mv) ^ ^ ^ gA
which follows at once from the equation
Forming next the two equations
ns
1/ =
A +
1
a'
2
(M +
•')«,
"(m
')/3
A
1
A'
"(m
2
«'))3
Km +
i')7
2/9},
these will be equivalent to a single equation if
(m + !/)• 8»= {0* + »')7  2/3{» + (/*  1/)* /8»,
that is, if
0* + !/)• S» = 0* + 1/)" (/S* + 7»)  4 (/i + v) i87  4 (/*•;  1) /S* ;
72 ANALYTICAL RESEARCHES CONNECTED WITH [114
or finally, if
which is in fact the case.
Writing ihe equations for
A 1 A 1
in the form
^ + T = ;^ \"^^" x = 7 r5(7 2w,
A (ji — vjps A {fivjPs^'
and substituting in
we have
U='^\[^^^(<'i+N+'iK)[^+^»{^+K)\,
^ ((/3 + 2«7 + 2^)f + (72«/3)i7 + (/9+2»7 + 4«^/8)?};
and consequently, multiplying by
we have
or collecting the different terms which enter into the equation
the result is
 1 V(7' + 2^/3')(7" + 20y8")/8'/8"l {( /3 + 2*y + 2^) f +(7  2»^) 17 + ( /8 + 2*/ + 4«^/9)r} = 0.
which, combined with the first equation written under the form
(of + ^, + yO*  «• [(f  ry + 17»] = 0,
determines the ratios of ^, 17, ^ that is, the values oi T + Z and YZ.
114] STiaNEB's EXTENSION OP MALPATTl's PROBLEM. 73
§6
The system of equations
(/+ 20 V^) + Vl (F+ Z ) +/YZ  \/bc ^/T+Y' 'JT+Zi = 0.
(g + 26 vIB) + VS(^ + X) +gZX  Vw \/l + Z* Vf+Z* = 0,
(A + 2^ V^) + V® (Z + F) + AXF  V^ Vr+X« VlTT' = 0,
where
on which depends the solution of Steiner's extension of Malfatti's problem, is at once
seen to belong to the class of equations treated of in the preceding section, and
we have ^ = 0, « = 0. The equations at the conclusion of the preceding section
become
a[(/+2^Vgr)f + Va, +/r] ^J(g+e VS) (h+e V®) Vig® {('^20f)^fy + Vi?}=o.
{(/+ 2^ va) I + v^, +/f}«  be {(f  (0* + »?'} = 0,
which may also be written
(VS® + Jf) (f + 5) + ( a Va + «7 V® + A ^18 + 25 Vl8®) (ij + 2^f )
 ^ Jig + e VlJ) (A + 5 V®) Vfi®! ((VI  25/) f /i; + '/^d = 0,
{/«+D + ^('? + 25f)}'^Kf0' + i?'} = 0.
Hence observing that
^+5Vii = ^(Vi8® + JF)(Vai8 + ^); A + dV® = ^(VJ8®+iF)(Va® + ffi);
oVa+AVS+5'V®+25VS®=5(VS®+jp),
and putting for a moment
X = ^ ^/(V^ + (5) (Vaa8 + 1^) V3B© ,
and therefore
V(^+ 5 VS)(A + 5 V«D) Vi8aD=(VJ8® + iF)x,
a n. 10
74 ANALYTICAL RESEARCHES CONNECTED WITH [114
the first equation divides by (^^^8CD + Jp), and the result is
Also, by an easy transformation, the second equation becomes
or putting
f+? + ^ (i? + 2^f ) = e,
the equations become
hence eliminating ^,
e  2x,o = 0,
or observing that
1 + ^ = ;^. (^^® + iF) (v^^a + ®) (>^ai8 + ?^),
and reducing, we obtain
*=.^=^
also B = 2X<I> gives
Suppose
then substituting
V3iar+ jF = '*' V3S® iF = «,. •■• «», = Ka,
P/r, \ Va + a/
114] STEINEE's extension of MALFATTfs PROBLEM. 75
that is,
Pfli
P.7/ V Va + a/
these may be written
L'^ + if 17 + i\r'c= 0,
where
P,7/ V Vo+a/
or since f, ly, ? are equal to 1, T+Z, YZ respectively,
1: Y + Z: YZ= MIT  MN : NL'  ITL : LM  L'M
^_46c^)/ V2V^ A / _ ^U^J+3(/+,Va).
^% ^ V/8,X >^ ^ Va + a/ V/8,7,
Also
whence
y ^^^ _ 2V2V^Tv^/l _ V2^^^^N / _ V^N _
^ ^ V)8,x A Va + a/
2V2 V^T^, ^^ /^tf , ^2 Va + a/ \ / V;; N
^ V "^ v;^, Jr va+v '
and by forming the analogous expressions for Z\X and ZX, X+F and XY, the
values of X, F, Z may be determined. But the equations in question simplify them
selves in a remarkable manner by the notation before alluded to.
10—2
76 ANALYTICAL RESEARCHES CONNECTED WITH [114
Suppose
these values give
iJ^ =f»g'h» + 2J>,
K* =f«g'h« + 2g»h» + 2h»f+2fV^^'.
Applying these results to the preceding formulae and forming for that purpose
the equations
» /s / la A 1. ^2 Va + o" J* Vo, f
2 V2 Va + a, V^,7, = 4gh, 73 =7s^. , — ^ = 7.
V^,7/ Vagh Vo + a, «/
ghA^ + ^ ITf = ( J«  gh) (f»  (g  h)*)  2gh (g  hy.
we have
if(F+^ + 2Z^ = 4(J*gh)(^l ^),
iTTZ + Z« = {(J'  gh) (f»  (g  h)*)  2gh (g  h)«} (1  ^) ;
the former of which, combined with the similar equations for Z\X and T + F, gives
for Z, F, Z the values to be presently stated, and these values will of course verify
the second equation and the corresponding equations for ZX and XY.
Recapitulating the preceding notation, if j? = 0, y = 0, z^O are the equations of the
given sections, t(;=»0 the equation of the polar plane of their point of intersection
with respect to the sur&ce,
CM5" + 6y* 4 c«* + 2^2? 4 2gr^ + 2fcry +/)«;» =
the equation of the sur&ce, ^, 38, ({D, $, <!Er, ^, iT as usual, and
114] steikeb's extension of malfatti's problem. 77
then the equations of the required sections are
{ax + hy + gz)X'^y V® + ^ ^^ + w V  op V 1 + Z« = 0,
X V® + y Va + (gx +/y + cz) Z+w^  cpVl + Z* = 0,
where X, F, ^ are to be determined by the following equations,
(/+ 25>/a)+ va (Y+z)hfYz  \/6c vrrF^viT^ = o,
(y + 2^ VS) +vs(.^+z)+^^zVcaV rr^ vi+z« = o,
(A + 25 V®)+ V^(Z + F) + AZF \/^ Vl+Z«Vrn^ = :
and the solution of which, putting
f=4'a>/vi5«^rS g=^©^/^^®affi, h=^®>/vai3?^, j^j~2<fmm.
is given by the equations
/irz = ^+( .f+g + h)«2(f+g + h)j,
ifF=?^ + ( fg + h)>2( fg + h)J,
ZZ = ?^^+( f + gh)>2( f+gh)/. C)
Instead of the direct but very tedious process by which these values of A", F, Z
have been obtained, we may substitute the following d posteriori verification.
We have
K*(l+X*) =4(f+g + h)'J'(l+^)(lj)(lJ).
2f,VlTF»VrT^ = 4 (f (jfh )) /• (l  j) \/l  $y/l  J, •
/r»(l + FZ) = * (l  j) {('^' «*»> i^(g^y)  2gh(g  h)",.
^(F+Z)2f» 2g« 2h« + 4/» = 4 (l ^) (J'gh).
Putting also
^ It IB periiapB worth noticing that the value of the quantity X previously made use of,
78
we have
ANALYTICAL RESEARCHES CONNECTED WITH
[114
(f. _ g. _ h« + ?^') IT* (1 + FZ)
= 4(lJ)(p(gh)')[(J'gh)(f«(gh)«)2gh(gh)«2gh^=^^5^]
4g' h'(gh)'(J»g h)
}■
hnj*gh)\
J'
K' {K(Y+Z)  2f« 2g' 2h' + 4/'}
= 4(i;)j{f(ghy)[(j»gh)((g+h).f*^?:)l*«^
Also, since
(f._(g.h).) + ((g+h).f^*^') = 4gh(:^>.
we have
(p ^ h' +^^\ K' (1 + YZ) + K* {K (Y + Z) it* 2^ 2h*}
= 4(l;J) (f«(gh).)2ghJ« (l ^.)(l ^,).
and the values obtained above give also
2gh \/l  ^\AJ. ^' ^ + ^' ^^1+^'
=4(i^)(f«(gh)02ghj«(if,)(i^;).
which shows that the relation between Y and Z is verified by the assumed values of
these quantities, and the other two equations are of course also verified. The solution
of the problem will be rendered more complete if the equations of the required sections
and of the auxiliary sections made use of in the geometrical construction are expressed
in terms of f, g, h, J.
§7.
First, to substitute in the equations of the required sections or resultors.
the first equation in the form
Writing
K'
the coeflBcient of x will be
aZa? + (AZ + V(!D)y + (grZ + V33)^ + VapVl + Z«w=0^
^ (
^j^l^'+if+s+^y^i'f^g^^)]^
114]
steiner's extension of malfatti's problem.
79
or, as it is convenient to write it.
1 +
^)r('.».h,4(i^){7p^^f...h^.
The coeflBcient of y is
^{(f.g.^.+^^X^'^*'^*'"^"''*^^''))
2Vi3
f 4 _ g4 _ h« + 2g»h' + 2hf + 2f g»  ^^^
or, after all reductions,
i} 4)«'^«^ (> « {.F^,*'«"^m^h
and similarly the coefficient of 2^ is
^(f.gH..^e)(?&t.(,.,.h).2^(f.g^h,)
_ f « _ g4 _ h« + 2g»h» + 2h»f + 2Pg'  ^^ j
or, after all reductions.
(,4)„.,.,..,^(,.h)_^^.,,,,.,,^2£^.b.,l
and the coefficient of u; is
(i + J)f(f+g+h)2Vzyiiy^i&yi^Vjt,.
Hence, forming the equation of the resultor in question, and by means of it those
of the other resultors, the equations of the resultors are
, _2fgh_
U(fVg + h)
 f + g 4 h  2/
)U'^'
+ U(f+g + h) + ^ g + h + 2J 2^^ j^\\ jjy
I 2fgh .,, . orf*g» + h»\ h /, h\
80 ANALYTICAL RESEARCHES CONNECTED WITH [114
+ 2 VF/^r^ /y/lTK y^T^ V^«, =
values which might be somewhat simplified by writing , t), ^, <o instead of
U''i)' Ui>' U'> V^^/'^V'"^^
and it may be also remarked, that the coefficients as well of these formulae as of those
which follow may be elegantly expressed in terms of the parts of a triangle having
f, g, h for its sides.
The equations of the separators are found by taking the differences two and two
of the equations of the resultors (this requires to be verified d posteriori) \ thus sub
tracting the third equation firom the second the result contains a constant fector,
j(f.(gh>')gh {^^g"^' ^ (f  (g  h)*) ((g+ by  p)}.
equivalent to
J(P(gh)^)ghV ^ V J^ J) (f«(ghy)gh
Rejecting the factor in question and forming the analogous two equations, the equa
tions of the separators are
^ (aa' + ...:? S*) = (aa + f^/8 + (Syy,
114] steiner's extension of malfatti's pkoblem. 81
and from the mode of formation of these equations it is obvious that the separators
have a line in common.
The equations of the determinators being x = 0, y = 0, e^Q, the equations of the
tactors are
v'S«V^y = 0, Vcf a;  V^« = 0, Vay V3©a;=0;
and if ax + fiy + yz + Bw = be the equation of the tactor touching
01 = 0, V®a!v'a« = and ViiyVSa! = 0,
the conditions of contact are
K
P
2 V15B ( VIS  1^) (aa" + . . .  8*) = ( VISB  1^) (a Va /3 VS) + 7 (® VS  1^ V^)l* ,
2 VI® ( va®  ffi) (aa* + . . .  s*) = Ws®  (5) (a vfi  7 V®) + ^ (?^ v®  jf va)l* .
whence
;;^V2Va3B(VaS^)(aa + ^/3 + €J7) =
(Vgoi  ^) vaa  (VIS  ^) vs/s + (ffi vs  jf va)7,
^V2Va®(va®ffi)(aa+^i8+ffi7) =
(Va®  ffi) Vgfa + (1^ V®  Jf VI) /3  (Vie  <!Ir) 7,
and putting for a moment
/i = vae  (5  V2"viiaD(Val€G),
V = Vi®  1^  72 vas (Vai8  1^);
after some reductions, and observing that the ratios only of the quantities a, fi, y, S
are material, we obtain
K
c. n.
va"
11
82 ANALYTICAL RBSBARCHB8 CONNECTED WITH [ll4
and it is easily seen also that the coordinates of the point of contact are
also
« = 0, y = v, z^fi, t^ =   ^ 5
'• ^('5). '=^('5)
Hence substituting and introducing throughout the quantities f, g, h, «/, also forming
the analogous equations, the equations of the tactors are
jp(f.+g.+h.)+(g+h)j(f.(gh)?^)j^.
+ 2>JK V gh (l  5) (l  j) (f  (gh)*) 'J~PW = 0,
+ 2V^ y hf (l  J) (l  j) (g"  (h  f )') J"^ = 0,
+ h. (f. + g.  h«) + (f + g) / (h»  (f  g)»  ^ijj*)! ^ ^
^•JKsJigil  j) (l  f) (h'(fg)') ^^^^> = 0
+
114] steiner's extension op malfatti's problem. 83
It is obvious, from the equations, that each separator passes through the point of
contact of a tactor and determinator, it consequently only remains to be shown that
each separator touches two tactors. Consider the tactor which has been represented
by aa; + /8y f 78r 4 Sw = 0, the unreduced values of the coefficients give
^aa«+...S» = ;;^(aa + f^5 + (E7) = iri
Represent for a moment the separator
h{^7)'U?)y'^'U^>"'
hy lx + my ■hnzh8W=: 0. Then putting ^P + ... — s» = Q', since
= ^{,(i4),(i^)(i5)(rg)(i^)(ii)}
the condition of contact becomes
D=r(fg)'+h(f+g)?^;
J
or, forming the value of □' and subetituting,
.(.^(r,„(i_5)(,4).(,.^>,(,.5)(,_5)
which may be verified without difficulty, and thus the construction for the resultors
is shown to be true.
11—2
84
ANALYTICAL RESEARCHES CONNECTED WITH
[114
§8
Several of the formulae of the preceding sections of this memoir apply to any
number of variables. Consider the surface (Le. hypersurface)
aai^ + bf + C2^+ 2fyz + 2gza) + 2hxy ... +pe* = 0,
and the section (i.e. hypersection)
where
(aX + A/i +5fv ...) a? + {h\ + bii +fv ...) y + (ff^ +/a* H ci' ••.) ^ ••• + ^ ""^ ^^ = 0,
V* = a\* + bfi* + cv" + 2//X1/ H 2gv\ + 2AV ...  K,
the condition of contact with any other section represented by a similar equation is
a\V + bfJLfi' + cw' +f(jiv + /jfp)hg (vXf + v\) + h (Kf/ + \» ... ±K = W,
where K is the determinant formed with the coeflScients a, 6, c, /, g, A, ... And con
sequently, by establishing all or any of the equations \ = \/^, /a = >/18, v = VCD, ...
we have the condition in order that the section in question may touch all or the
corresponding sections of the sections a? = 0, y = 0, z = Of ...
Let tt be the number of the variables x, y, z... , then K^"^ =.
G :ff (S^
also
Z«» {(a\ + hfM + gv ...) x + (h\ + bfi +fa ...) y + {9>^ ^ff^ + op ...) z ...]
X
y
2
X a
1^
ffi
A* ?^
IS
JF
I* Ci
iF
®
whence also
Z«*(V» + ^ = 
X /»
V ...
or /ir"*v> = 
X a 1^
e
^ ?^ 53
JF
1 " ffi iF
1 •
®
1
X
/*
V
X
^
?^
®
/*
n
19
iF
»
€r
iF
€
114]
STEINBBS EXTENSION OF MALFATTIS PBOBLEM.
85
and the equation of the section in question becomes
X
y
z ...
+ ir*«>vj>v
\ &
n
&
t* n
i3
JF
V <&
•
•
iF
e
1 \
iF
also the condition of contact with the corresponding section is
Tl
X
/*
V
V
»
?^
dr
1^
13
iF
l/
eEr
iF
®
V
1 X
/*
V ...
X a
1^
<!1t
M n
13
iF
•
•
•
iF
®
V ...
iF
^
« = 0,
V
1 X'
/*'
y'... 1
X' a
^
dr
/*' »
as
iF
v' ®
•
•
•
iF
e
In particular the equation of the sections which touches all the sections x^O, y
«=0, ..., is
= 0,
X y z
V® <S jp CD
+
if jiii V  p yn
1 Via vs v®... U=o.
va a ?^ ffi
v^ f^ 18 iF
V^ (ffir iF ffi
Again, the equations of the section touching y = 0, ^ = 0,... and the sections touching
«sO, ^ = 0, ... are
> = 0,
tc
y
z
X
a
?^
dr
v13
?^
33
iF
•
•
•
df
iF
e
a;
y
2
va
a
?^
dr
^
?^
13
iF
V®
df
iF
®
+z»«wj3 7
1
X
V®
•
•
•
X Vi3 V®...
a ?^ dr
1^ 55 iF
di JF ®
+ Jfi«Wp>/^
1
V®
•
•
•
va" /* ^ 
a ?^ dr
1^ 3D iF
ffi JF ®
1< = o,
86
ANALYTICAL RESEABCHE8, &C.
[114
and the condition of contact of these two sections is
+ 1
X
Vi3
m
sa
1^
/*
m
13
V®
&
iF
V<ZD...
JF
= V_"
1
X
^23 V®...
\
^
1^ cs
V39
1^
38 iF
V@
<!1t
iF ®
y
1 va
18 iF
iF ®
It would seem from the appearance of these equations that there should be some
simpler method of obtaining the solution than the method employed in the previous
part of this memoir.
115]
87
115.
NOTE ON THE PORISM OF THE INANDCIRCUMSCRIBED
POLYGON.
[From the Philosophical Magazine, voL VL (1853), pp. 99 — 102.]
The equation of a conic passing through the points of intersection of the conies
17=0, F=0
is of the form
wU+V=0,
where i(; is an arbitrary parameter. Suppose that the conic touches a given line, we
have for the determination o( w a, quadratic equation, the roots of which may be
considered as parameters for determining the line in question. Let one of the values
of w he considered as equal to a given constant k, the line is always a tangent to the
conic
kU+V = 0;
and taking w^^p for the other value o{ w, p is a parameter determining the particular
tangent, or, what is the same thing, the point of contact of this tangent.
Suppose the tangent meets the conic U = (which is of course the conic corre
sponding to w = co)m the points P, P^, and let 0, oo be the parameters of the point
P, and ff, 00 the parameters of the point P'. It follows from my "Note on the
Geometrical representation of the Integral Jdxi V(aj + a) (a? + 6) (x + c)," [113] (*) and
from the theory of invariants, that if nf represent the "Discriminant" of ^U + V
^ I take the opportonity of oorreoting an obvious error in the note in question, viz. a'+&'+c*2&c2eii2a5
is throughout written instead of (what the expression should be) 6*c'+c'a*+a*&'2a'&e2&>ea2c^. [This
coiraotion is made, ante p. 55.]
88 NOTE ON THE PORISM OF THE INANDCIRCUMSCBIBED POLYGON. [ll5
(I now use the term discriminant in the same sense in which determinant is sometimes
used, viz. the discriminant of a quadratic function cue* + iy* + cg^' + %fyz + 2gzx + 2hxy
or (a, 6, c, /, g, h) {x, y, zy, is the determinant k = abc — of* — 65^ — ch* + 2fgh), and if
•^ 00
df
then the following theorem is true, viz.
"If {0y 00), (ffy 00) are the parameters of the points P, P' in which the conic
U=0 is intersected by the tangent, the parameter of which is p, of the conic
kU+V=0, then the equations
n^ ^npUk,
n^=np+nA:,
determine the parameters 6, ff of the points in question." And again, —
"If the variable parameters ^, ff are connected by the equation
n^n^=2nit,
then the line Ff will be a tangent to the conic fcCr+F=0." Whence, also, —
" If the sides of a triangle inscribed in the conic 17= touch the conies
k I7+F=0,
it'I7+F=0,
rr7+F=o,
then the equation
nifc+nifc'+nr=o
must hold good between the parameters fc, k\ VT
And, conversely, when this equation holds good, there are an infinite number of
triangles inscribed in the conic [7=0, and the sides of which touch the three conies ;
and similarly for a polygon of any number of sidea
The algebraical equivalent of the transcendental equation last written down is
1, k , Vn*
1, V, VqF
= 0;
1, y\ VqF
let it be required to find what this becomes when k^V ^kf' ^0^ we have
115] NOTE ON THE P0RI8M OF THE INANDCIRCUMSCRIBED POLYGON. 89
and substituting these values, the determinant divides by
1, k\ ifc'»
1, r, r«
the quotient being composed of the constant term C, and terms multiplied by k, k\ k" ;
writing, therefore, fc = A/ = fc" = 0, we have (7=0 for the condition that there may be
inscribed in the conic Z7 = an infinity of triangles circumscribed about the conic
F=0; C is of course the coeflBcient of f" in Vnf> ie. in the square root of the
discriminant of f 17"+ F; and since precisely the same reasoning applies to a polygon
of any number of sides, —
Theorem. The condition that there may be inscribed in the conic CT^O an
infinity of 7»gons circumscribed about the conic F = 0, is that the coefficient of f*"' in
the development in ascending powers of f of the square root of the discriminant of
f[7+ F vanishes. [This and the theorem p. 90 are erroneous, see po»^, 116].
It is perhaps worth noticing that w = 2, i. e. the case where the polygon degene
rates into two coincident chords, is a case of exception. This is easily explained.
In particular, the condition that there may be in the conic ^
flwr»+6y*cz'=0
an infinity of »gons circumscribed about the conic
ic*+ y'+ ^' = 0,
is that the coeflScient of 1^^ in the development in ascending powers of f of
\/(l + a^)(l+6^)(lhcf)
vanishes ; or, developing each factor, the coefficient of f**~* in
(l + iafJa«f»hT'ffa'??fia'^'+&c.)(l + i6f&c.)(l + ic^&c.)
vanishea
Thus, for a triangle this condition is
a^ + 6» + c»  26c  2ca  2a6 = ;
for a quadrangle it is
a» + 6* + c*  6c« 6»c  ca»  c^a  a6»  a'6 + 2a6c = 0,
which may also be written
(6 + c  a) (c + a — 6) (a + 6  c) = ;
and similarly for a pentagon, &c.
' I have, in order to present this result in the simplest form, purposely used a notation different from
^t of the note above referred to, the quantities ax>+&y*+c;e' and o^hy'^^'Z^ being, in fact, interchanged.
c. n. 12
90 NOTB ON THE PORISM OF THE INANDOIRCUMSCRIBED POLYGON. [llo
Suppose the conies reduce themselves to circles, or write
i2 is of course the radius of the circumscribed circle, r the radius of the inscribed
circle, and a the distance between the centres. Then
f[7+F=(f+l, f+1, fi?^i + a«, 0, a, 0)(x, y, 1)«,
and the discriminant is therefore
Hence,
Theorem. The condition that there may be inscribed in the circle a^ + y*  ii*^ =
an infinity of ngons circumscribed about the circle (j?— a)*+ y* — r* = 0, is that the
coefficient of f"""* in the development in ascending powers of f of
may vanish.
Now
(A +Bf + 6'?)* = VJ l + i5 1 + (i^C  i5«) f. + ...} .
or the quantity to be considered is the coefficient of f^^ in
(1 +*f4p) {i + ifi+ (i^c 45«);+ ...,.
where, of course,
In particular, in the case of a triangle we have, equating to zero the coefficient
of p,
or substituting the values of A, B, C,
(a»  i?)«  4r»i? = 0,
that is
(a» i?+ 2iJr)(a»i? 2iJ7) = ;
the factor which corresponds to the proper geometrical solution of the question is
a»i? + 2iJr=0,
Euler's wellknown relation between the radii of the circles inscribed and circumscribed
in and about a triangle, and the distance between the centres. I shall not now discuss
the meaning of the other factor, or attempt to verify the formulae which have been
given by Fuss, Steiner and Richelot, for the case of a polygon of 4, 5, 6, 7, 8, 9, 12,
and 16 sides. See Steiner, CreUe, t. n. [1827] p. 289, Jacobi, t. in. [1828] p. 376 :
Richelot, t. v. [1830] p. 250; and t. xxxviii. [1849] p. 353.
2 Stone Buildings, July 9, 1853.
116]
91
116.
COREECTION OF TWO THEOREMS RELATING TO THE PORISM
OF THE INANDCIRCUMSCRIBED POLYGON.
[From the Philosophical Magazine, vol. VL (1853), pp. 376 — 377 ]
The two theorems in my " Note on the Porism of the inandcircumscribed Polygon "
(see August Number), [115], are erroneous, the mistake arising from my having in
advertently assumed a wrong formulae for the addition of elliptic integrals. The first
of the two theorems (which, in fact, includes the other as a particular case) should be as
follows : —
Theobem. The condition that there may be inscribed in the conic 17 = an
infinity of ngons circumscribed about the conic F=0, depends upon the development
in ascending powers of f of the square root of the discriminant of fZ7+ F; viz. if
this square root be
then for n = 3, 5, 7, &c. respectively, the conditions are
C=0,
C, D
D, E
= 0,
C, D, E
D, E. F
E, F, Q
*nd for n = 4, 6, 8, &c respectively, the conditions are
= 0, &C. ;
ID 1=0,
D, E
E, F
= 0,
= 0, &c.
D, E. F
D, F,
F, G, H
^6 examples require no correction; since for the triangle and the quadrilateral
'^spectively, the conditions are (as in the erroneous theorem) (7 = 0, Z) = 0.
12—2
92 CORRECTION OF TWO THEOREMS RELATING TO THE PORISM, &C. [llG
The second theorem gives the condition in the case where the conies are replaced
by the circles ^r* + y* — i? = and (a? — a)* + y' — ^ = 0, the discriminant being in this case
 (1 + f ) {r» + f (r« + i?  aO + f^iP}.
As a very simple example, suppose that the circles are concentric, or assume
a = ; the square root of the discriminant is here
(l+f)\/r» + i?f;
and putting for shortness ^ = «» we may write
il+5f+...=(l+f)VrTif,
that is, ^=1, fi = ia + l. C = ia» + ia», 2) = tV«'*«"> ^ = TfF«* + TV«'> &c. ;
thus in the case of the pentagon,
C^D» = j^a*{(a4)(5a8)4(a2)«l
= tAi «*(«* 12a + 16),
and the required condition therefore is
a»12a + 16 = 0.
It is clear that, in the case in question,
^ = cos36° = i(^5 + l),
that is,  = V51, or (12 + r)«5r«= 0,
viz. (Va + 1)"  5 = 0, or a + 2 Va 4 = 0,
the rational form of which is
(^ 12a + 16 = 0,
and we have thus a verification of the theorem for this particular case.
2 Stone Buildings, Oct. 10, 1853.
117]
93
117.
NOTE ON THE INTEGRAL Idx^Jim^x) (x + a){x + b){x + c).
[From the Philosophical Magazine, vol. vi. (1853), pp. 103 — 105.]
If in the formubB of my " Note on the Porism of the inandcircumscribed Polygon/'
[115], it is assumed that
and if a new parameter cd connected with the paramieter w by the equation
com
w=
m — G)
be made use of instead of w, then
and thus the equation wU+V=Oy viz. the equation
G)(d5» + y" + 2^) + aa:' + 6y« + C2» = 0,
is precisely of the same form as that considered in my "Note on the Geometrical
Representation of the Integral Idx = V(a: + «)(« + 6) {x + c)," [113.] Moreover, introducing
instead of f a quantity 17, such that
then
VDf "Jimri) (a + 17) (6 + 17) (c + 17)'
94 NOTE ON THE INTEGRAL j dxi J{m'X) (x + o) (x + b) (x + c). [ll7
Also f = 00 gives 17 = m, the integral to be considered is therefore
11,17=1 .  — ;
Jm v(mi7)(a + i7)(D + i7)(c + i7)
i.e. if in the paper last referred to the parameter 00 had been throughout replaced
by the parameter m, the integral
drj
ni7=f j=
V(a 4 17) (6 + 17) (c + 17)
would have had to be replaced by the integral 11,17. It is, I think, worth while to
reproduce for this more general case a portion of the investigations of the paper in
question, for the sake of exhibiting the rational and integral form of the algebraical
equation corresponding to the transcendental equation ±Tl,k±TI,p ±11,0=^0, Consider
the point f, 17, f on the conic m(j:' + y' + ^) + aaj"f fty" + C£i" = 0, the equation of the
tangent at thife point is
(m + a) f a? + (m + 6) i7y + (m + c) 5? = ;
and if ^ be the other parameter of this line, then the line touches
^ (a;» f y " + ^) + OA" + 6y> + c^ = ;
or we have
d + a "^ ^ + 6 ^ + c ~ '
and combining this with
(m + a)f" + (m + 6)i7« + (m + c)f = 0,
we have
f : 17 : g'= ^ b "C 'Ja^d "Jb km ^c+m
: ^{c'a)'Jb + 0^c {•m'Ja + m
: V(a 'b)'^C'\0^a + m^b'km
for the coordinates of the point P. Substituting these for x, y, z in the equation of
the line PP" (the parameters of which are p, k\ viz. in
a?V6c v^(a4 A;)(a+p) + yVcaV(6 + A;)(6+p) + 2:Va6Vc + fcVc+;)=0,
we have
'Ja + m V6 +
m
vc + m
117] NOTE ON THE INTEGRAL I dx^ J{m'X) (x + o) (x + b) (x + c). 95
which is to be replaced by
(a+p)(ah^)(a + g)^
6 + m
These equations give, omitting the common factor (a + wi) (6 + m) (c + m),
+ {abc {kd\0p+ kp) \pk6 {bc{ca + a6)j,
{■m[''aJbc pkO^ (bc + cakab) (/> + *+ 5) + (*^ + ^[p +;>*)(« + 6 + c)}
+ {oftc (p + ^* + ^)  pk0 (a + 6 + c)} ,
+ m {{be {ca^ ab)  {k6 + 6!p +#)}
+ ahc+pkO;
and substituting in 4X*. ft' — (2X/a)* = 0, we have the relation required. To verify that
the equation so obtained is in fact the algebraical equivalent of the transcendental
equation, it is only necessary to remark, that the values of \\ /i' are unaltered, and
that of \fi only changes its sign when a, 6, c, m and />, k, 0^ —m are interchanged ;
and so this change will not affect the equation obtained by substituting in the equation
4X' . ^' — (2X/a)' = 0. Hence precisely the same equation would be obtained by eliminating
L, M from
(ifc + a) (ifc + 6) (ifc + c) = (Z + Jfit)» (m  A),
(p + a)(p + 6)Cp + c) = (Z + ifi>)*(mp),
(^ + a)(5 + 6)(d+c) = (Z + Jftf)«(dj9);
or, putting (Z + Mk) (m— k)='a + ^k + ^h?, by eliminating a, ^, 7 from
(m  k){k + a) (k + 6) (i + c) = (a + /3A + yk'Y,
(mp)(p + a)(/> + 6)(/> + c) = (a + /3p + 7p«)«,
(m5)(5+a)(5+6)((9+c) = (a + /9^ + 7^)^
= (a + ^m + 7m')«,
which by Abel's theorem show that p, i, ^ are connected by the transcendental equation
above mentioned.
2 Stone BuildingSy July 9, 1853.
96
[118
118.
ON THE , HAEMONIC RELATION OF TWO LINES OR TWO
POINTS.
[From the Philosophical Magaziney vol. VI. (1863), pp. 103 — 107.]
The "harmonic relation of a point and line with respect to a triangle'* is well
known and understood ^ ; but the analogous relation between two lines with respect to
a quadrilateral, or between two points with respect to a quadrangle, is not, I think,
sufficiently singled out from the mass of geometrical theorems so as to be recognized
when implicitly occurring in the course of an investigation. The relation in question,
or some particular case of it, is of frequent occurrence in the Traits des ProprOtis
Projectives, [Paris, 1822], and is, in fact, there substantially demonstrated (see No. 163);
and an explicit statement of the theorem is given by M. Steiner, Lehrsatze 24 and 25,
Crelle, t. xiil [1835] p, 212 (a demonstration is given, t. xix. [1839] p. 227). The
theorem containing the relation in question may be thus stated.
Theorem of the harmonic relation of two lines with respect to a qvadrUaieral. " If
on each of the three diagonals of a quadriUiteral there be taken two points harmonically
related with respect to the angles upon this diagonal, then if three of the points lie
in a liney the other three points will also lie in a line^* — the two lines are said to
be harmonically related with respect to the quadrilateral.
It may be as well to exhibit this relation in a somewhat different form. The
three diagonals of the quadrilateral form a triangle, the sides of which contain the
six angles of the quadrilateral; and considering three only of these six angles (one
angle on each side), these three angles are points which either lie in a line, or else
^ The relation to which I refer is contained in the theorem, *'If on each side of a triangU there be
taken two points harmonicaUy related with respect to the angles on this side, then if three of these points
lie in a line, the lines joining the other three points with the opposite angles of the triangle meet in a
point,** — the line and point are said to be harmonically related with respect to the triangle.
118] ON THE HARMONIC RELATION OF TWO LINES OR TWO POINTS. 97
are such that the lines joining them with the opposite angles of the triangle meet in
a point. Each of these points is, with respect to the involution formed by the two
angles of the triangle, and the two points harmonically related thereto, a double point;
and we have thus the following theorem of the harmonic relation of two lines to
a triangle and line, or else to a triangle and point.
Theorem. '' If on the sides of a triangle there be taken three points, which either
lie in a line, or else are such that the lines joining them with the opposite angles
of a triangle meet in a point; and if on each side of the triangle there be taken
two points, forming with the two angles on the same side an involution having the
firstmentioned point on the same side for a double point ; then if three of the six
points lie in a line, the other three of the six points will also lie in a line," — the
two lines are said to be harmonically related to the triangle and line, or (as the case
may be) to the triangle and point.
The theorems with respect to the harmonic relation of two points' are of course
the reciprocals of those with respect to the harmonic relation of two lines, and do
not need to be separately stated.
The preceding theorems are useful in (among other geometrical investigations) the
porism of the inanddrcumscribed polygon.
2 Stone Buildings, July 9, 1863.
C. n.
13
98
[119
119.
ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL.
[From the Philosophical Magnziney vol. vi. (1853), pp. 182—185.]
The theorem to which I refer is remarkable for the extreme simplicity of its
demonstration. Let it be required to expand the factorial x—a x — h x^c ,.. in the
form
x — a x — x — y...^ Bx" a a?^... + Oa? — a ... + D ... &c.
We have first
a: — a = ar — a + a — a;
multiply the two sides of this by a: — 6 ; but in multiplying by this factor the term
.'/; — a, write the factor in the form a? — ^ + iS — 6 ; and in multiplying the term a — a,
write the factor in the form x — a + a — b; the result is obviously
x — ax— h^x — ax — fi +(a — a + /8 — 6)a? — a +a— aa— 6;
multiply this by a; — c, this factor being in multiplying the quantity on the righthand
side written successively under the forms a: — 7 + 7 — c, a?— /8I/8 — c, a — a^a — c] the
result is
X — a x^b X — c= x — ax — fix — y
+ (a — ahyS — 6 + 7 — c)a: — aa: — /8
+ (a — a a— 6 + aa /S — c^fi — b jS — c) x  a
119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 99
which may be thus written,
(x — a) (a? — 6) (ar — c) =
Consider, for instance,
La, 6, cJ,
then, paying attention in the first instance to the Greek letters only, it is clear that
the terms on the second side contain the combinations two and two, with repetitions,
of the Greek letters a, ^, and these letters appear in each tenn in the alphabetical
order. Each such combination may therefore be considered as derived from the primitive
combination a, a by a change of one or both of the as into ^; and if we take
(instead of the mere combination a, a) the complete first term a — a a — 6, and
simultaneously with the change of the a of either of the factors into ^ make a similar
change in the Latin letter of the factor, we derive from the first term the other terms
of the expression on the righthand side of the expression. It is proper also to
remark, that, pa3dng attention to the Latin letters only, the different terms contain
all the combinations two and two, without repetitions, of the letters a, 6, c. The same
reasoning will show that
x — ax — bx — cx — d== x^a x — x — y x — 8
La, 6, c, aJj
la, b, c, dJ,
[a. /8 
+ x — a
La, 6, c, dJ,
La, b , c, dJt
where, for instance,
La, b, c, dit +(aa)(ab)(^d)
+ (aa)(/8c)(/9d)
+ (/96)(/3c)(/9d),&e.
13—2
100 ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. [119
It is of course easy, by the use of subscript letters and signs of summation, to
present the preceding theorem under a more condensed form ; thus writing
LC&i U] ... Of , . . (lf.^gj f ^1 ^ )
where k,, Avi, ...&o form a decreasing series (equality of successive terms not excluded)
of numbers out of the system r, r — 1, ...3, 2, 1; the theorem may be written in the
form
X
but I think that a more definite idea of the theorem is obtained through the notation
first made use o£ It is clear that the above theorem includes the binomial theorem
for positive integers, the corresponding theorem for an ordinary factorial, and a variety
of other theorems relating to combinationa
Thus, for instance, if Gg(ai,...ap) denote the combinations of a,, ... Op, q and q
together without repetitions, and ^g (Oi, ... Op) denote the combinations of a,, ... Op,
q and q together with repetitions, then making all the a's vanish,
and therefore
a? tti ... ajOp = ^^o()' C'gCoi* ••• ap)«*^r
(X  a)P = SfoH' C, (a, a . . . plures) a^ = ^ S H' H a^ ^^.
the ordinary binomial theorem for a positive and integral index p.
So making all the a's vanish,
a?** = /S^^off9(«i ••• «p«+i) ^^ *i ^"" ^ •••*■" "p«
If m be any integer less than p^ the coefl&cient of x^ on the righthand side
must vanish, that is, we must have identically
So also
ffti . . . ci^«^+i I
Cp^((h, a,, ... Op) = Sqo ()«Ci_^M»(«i» «» ••• «P«) [^ ^J^
Suppose
01 = 0, a, = l ...ap=2) — 1; ax=A, o, = i:— 1,... ap«fc2)+ 1,
then
[:':..^J.[":!.:.!:;:at^t*''^'"^^'''"^
and hence
the binomial theorem for factorials.
119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 101
A preceding formula gives at once the theorem
It may be as well to remark^ with reference to a demonstration frequently given
of the binomial theorem, that in whatever way the binomial theorem is demonstrated
for integer positive indices, it follows from what has preceded that it is quite as easy
to demonstrate the corresponding theorem for the factorial [m]^. But the theorem
being true for the factorial [m]^, it is at once seen that the product of the series
for (l+xy* and (1 +«?)** is identical with the series for (l+a:)*^+^ and thus it becomes
unnecessary to employ for the purpose of proving this identity the socalled principle
of the permanence of equivalent forms ; a principle which however, in the case in
question, may legitimately be employed.
102
[120
120.
NOTE ON A GENERALIZATION OF A BINOMIAL THEOREM.
[From the Philosophical Magazine, vol. vi. (1853), p. 185.]
The formula {CrelUy t. i. [1826] p. 367) for the development of the binomial {x + a)*',
but which is there presented in a form which does not put in evidence the law of
the coefficients, is substantially equivalent to the theorem given by me as one of the
Senate House Problems in the year 1851, and which is as follows : —
"If {a 4^8 + 7...)'' denote the expansion of (a 4^ + 7... )p, retaining those terms
Na^ff^rfh^ ,,. only in which 6 + c + d... is not greater than /) — 1, cfrff.. is not greater
than p — 2, &c., then
ar** = 1 (a: + o)**
 "^"7'2W^^ {«h/3h7N^ia4/3h7 + 8).
+ &c.'
>i
The theorem is, I think, one of some interest.
121]
103
121.
NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES.
[From the Philosophical Magazine^ vol. vi. (1853), p. 259.]
The following question was suggested to me, either by some of Prof Boole's
memoirs on the subject of probabilities, or in conversation with him, I forget which ;
it seems to me a good instance of the class of questions to which it belongs.
Given the probability a that a cause A will act, and the probability p that A
acting the effect will happen; also the probability /8 that a cause B will su^t, and the
probability q that B acting the effect will happen; required the total probability of
the effect.
As an instance of the precise case contemplated, take the following: say a day is
called windy if there is at least w of wind, and a day is called rainy if there is at
least r of rain, and a day is called stormy if there is at least W of wind, or if
there is at least R of rain. The day may therefore be stormy because of there being
at least W of wind, or because of there being at least R of rain, or on both 8w;counts ;
but if there is less than W of wind and less than R of rain, the day will not be
stormy. Then a is the probability that a day chosen at random will be windy, p the
probability that a windy day chosen at random will be stormy, /8 the probability that
a day chosen at random will be rainy, q the probability that a rainy day chosen at
random will be stormy. The quantities X, fi introduced in the solution of the question
mean in this psurticular instance, \ the probability that a windy day chosen at random
will be stormy by reason of the quantity of wind, or in other words, that there will
be at least W of wind; fi the probability that a rainy day chosen at random will
be stormy by reason of the quantity of rain, or in other words, that there will be at
least R of rain.
104 NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES. [l21
The sense of the terms being clearly understood, the problem presents of course
no difficulty. Let X be the probability that the cause A acting will act efficaciously ;
fi the probability that the cause B acting will act efficaciously; then
P = \ + (1X)aa^.
g = /[A + (l/Lt)aX,
which determine \, fi] and the total probability p of the effect is given by
p = Xa + ^  X/itt/S ;
suppose, for instance, a=l, then
;? = X + (lX)/i^, g = /Lt+ XX/Li, p = X + /Lt^Xft/8,
that is, p = p, for p is in this case the probability that (acting a cause which is
certain to act) the effect will happen, or what is the same thing, p is the probability
that the effect will happen.
Machynlleth, August 16, 1853.
122]
105
122.
ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE
OF THE SECOND ORDER INTO ITSELF.
[From the Philosophical Magazine, vol. vi. (1853), pp. 326—333.]
The following theorems in plane geometry, relating to polygons of any number
(odd or even) of sides, are well known.
" If there be a polygon of (m + 1) sides inscribed in a conic, and m of the
sides pass through given points, the (m + l)th side will envelope a conic having double
coutact with the given conic." And " If there be a polygon of (m + 1) sides inscribed
^ a conic, and m of the sides touch conies having double contact with the given
^nic, the {m + l)th side will envelope a conic having double contact with the given
conic.*' The second theorem of course includes the first, but I state the two separately
for the sake of comparison vdth what follows.
As regards the corresponding theory in geometry of three dimensions, Sir W. Hamilton
given a theorem relating to polygons of an odd number of sides, which may be
thus stated: "If there be a polygon of (2m + 1) sides inscribed in a surface of the
^^ud order, and 2m of the sides pass through given points, the (2m + l)th side will
constantly touch two sur&ces of the second order, each of them intersecting the given
8urfece of the second order in the same four lines ^"
^ See Phil, Mag. vol xxzt. [1S49] p. 200. The form in which the theorem is exhibited bj Sir W. Hamilton
u somewhat different ; the surface containing the angles is considered as being an ellipsoid, and the two sorfaoes
^'^'i^ hj the last or (2]ii + l)th side of the polygon are spoken of as being an ellipsoid, and a hjperboloid of
two sheets, having respeotivelj doable contact with the given ellipsoid : the contact is, in fact, a qoadraple con
^ ^ the same four points ; real as regards two of them in the case of the ellipsoid, and as regards the other
^ ift the case of the hyperboloid of two sheets ; and a qoadraple contact is the coincidence of foar generating
^^>Mi belonging two and two to the two series of generating lines, these generating lines being of coarse (in the
^Me Qoniidered by Sir W. Hamilton) all of them imaginary.
c. n. 14
106 ON THE HOMOQRAPHIC TBANSFOBMATION OF [122
The entire theory depends upon Tvhat may be termed the transformation of a
sur£BM^ of the second order into itself, or analytically, upon the transformation of a
quadratic form of four indeterminates into itself I use for shortness the term trans^
formation simply; but this is to be understood as meaning a homographic transformation,
or in analytic language, a transformation by means of linear substitutions. It will
be convenient to remark at the outset, that if two points of a surface of the second
order have the relation contemplated in the data of Sir W. Hamilton's theorem (viz.
if the line joining the two points pass through a fixed point), the transformation is,
using the language of the Recherchss Arithm^tiqtiea, an improper one, but that the
relation contemplated in the conclusion of the theorem (viz. that of two points of a
surface of the second order, connected by a line touching two surfaces of the second
order each of them intersecting the given sur&ce of the second order in the same
four lines) depends upon a proper transformation; and that the circumstance that an
even number of improper transformations is required in order to make a proper trans
formation (that this circumstance, I say), is the reason why the theorem applies to
polygons in which an even number of sides pass through fixed points, that is, to
polygons of an odd number of sides.
Consider, in the first place, two points of a sur£ace of the second order such that
the line joining them passes through a given point. Let a, y, z, w he current
coordinates S and let the equation of the surface be
(a,. ..)(«, y, z, wy = 0,
and take for the coordinates of the two points on the surface Xi, yi, Zi, v\ and
^i> Vti ^s> ^1} ai^d for the coordinates of the fixed point a, /3, 7, S. Write for shortness
(a, ...)(«, /S, 7, S)«=p,
(a, ...)(«, ^, 7, S)(a?i, yi, z^, Wi) = qu
then the coordinates ^, y,, z^, w^ are determined by the very simple formulsB
2ce
27
P
2S
^ Strictly speaking, it is the ratios of these qoantities, e.g. « : tr, y : 19, z : w, whioh are the ooordinates, and
consequently, even when the point is given, the values «, y, «, to are essentially indeterminate to a factor prh.
So that in assu mi ng that a point is given, we should write xiyiz: w=a : P'.yiH; and that when a point is
obtained as the result of an analytical process, the conclusion is necessarily of the form just mentioned : but
when this is once understood, the language of the text may be properly employed. It may be proper to explain
here a notation made use of in the text: taking for greater simplicity the case of forms of two variables,
(I, m) {x^ y) means te+my ; (a, 6, e) («, y)« means a«*+26xy+cy«; {a,b,e) ft, iy) (ar, y) means a^x + h{fy + rtx) + e7fy.
The system of coefficients may frequently be indicated by a single coefficient only : thus in the text (a, ...) («, y, f , ir)^
stands for the most general quadratic ftmotion of four variables.
122]
A SUBFACE OF THE SEOOND ORDER INTO ITSELF.
107
In &ct, these values satisfy identically the equations
«i, y%, ^, w%
0,
«i, Vu Sk, Wi
a, 0, y, S
that is, the point (xt, y%, z^, w^ will be a point in the line joining {Xi, j/i, z^ Wi)
and (a, fi, y, S). Moreover,
(a, ...)(«!, yi» ^1, w«)*= (a» ...)(«^i. Vu ^» ^y
^ (a, ...)(«, A 7, S)(«i, yi, Zu Wi)
(a, ...) («i, yi, ir^, t^i)«  ^' ?i + % i>,
that is,
80 that d?, yi, Zi, Wi being a point on the surface, a?„ y,, z^, w^ will be so too. The
equation just found may be considered as expressing that the linear equations are a
transformation of the quadratic form (a, ...)(a;, y, z, vif into itself If in the system
of linear equations the coefficients on the righthand side were arranged squarewise,
and the determinant formed by these quantities calculated, it would be found that
the value of this determinant is —1. The transformation is on this account said to
be improper. If in a system of linear equations for the transformation of the form
mto itself the determinant (which is necessarily h 1 or else — 1) be +1, the trans
formation is in this case said to be proper.
We have next to investigate the theory of the proper transformations of a quadratic
form of four indeterminates into itself This might be done for the absolutely general
form by means of the theory recently established by M. Hermite, but it will be
sufficient for the present purpose to consider the system of equations for the trans
f(Mination of the form aj*hy' + 2^ + ti;* into itself given by me some years since. (Crdle,
voL xxxn. [1846] p. 119, [52] Q).
I proceed to establish (by M. Hermite's method) the formulae for the particular
case in question. The thing required is to find ^, y„ z^, w^ linear functions of
"hf Vu ^t ^u such that
^' + y«* + z^^ I 1^,« = a?i* + yi« + ^1* + Wi*.
Write
flr,+«i=2f, yi + y, = 2i7, Zi + z^:=2Z ii;i + w, = 2o);
^ It is a jjingnlar instanoe of the way in which different theories connect themselves together, that the
toannls in qoeetion were gooeralizations of Baler's formolaB for the rotation of a solid body, and also are
loaaavlm which reappear in the theory of qaatemions ; the general formolas cannot be established by any obyions
ggpsraliaation of the theory of qaatemions.
14—2
108 ON THE HOMOGRAPHIC TRAJ^SFOEMATION OF [122
then putting a:, = 2f — a?i, &c., the proposed equation will be satisfied if only
f" + if + (7+ «*= f^ + Wi + f^i + a)Wi,
which will obviously be the case if
Zi = /Ltf — Xiy + ? + ccD ,
Wi = — af — 617 — c(^ + a ,
where X, /li^ 1^, a, b, c are arbitrary.
Write for shortness
a\ + bfjL + cv^il>, l + X« + /Lt« + i/» + a« + 6' + (^ + 0« = A;,
then we have
A:f = (l +X*+ 6"+ (^)a^ + (XMy a5c0)yi + (i'X. + /ica + 6^)^i + (6i;c/ia\^)wi,
A:(;' = (i'X/Ltca6^)a?i + (AM/ + X — 6c +a^)yi+(l +i^ + a*+fe» )?!+ (aft—ftX. 01/^)1^1,
A;u;= (61/  c/i+ a +\^) a?i + (cX. ay+ 6 +/A^)yi+ (a^ 6*'+ c + 1/^) ^1 + (1 + X* +/Lt» + i^) i^/i ;
and fix>m these we obtain
Auji = (1 + X» + 6« + (^  /Lt* 1/' a'  ^)a?i + 2 (X/Lii/ oft c<^)yi + 2 (vX. + Mca + 6^) ^1
+ 2 (6v — c/Li — a— X^) Wi,
A:ya = 2(X/Lt + ya6 + c0)a?i + (l + M* + (^ + a«i^X«fe«<^)yi42(/[AyX6ca^)^^
+ 2 (cX — ai/ — 6 — /Lt^) w/j ,
A:?a = 2(i/X/Ltca6^)a?i + 2(/Ay + X6c + a^)yi + (l + i/» + a» + 6*X«/Lt*c>«^)^^
+ 2 (a/Li — 6X — c — y^) Wi ,
Arwj = 2 (61/ — c/Li + a + X^) a?i + 2 (cX  ai/ + 6 + ft^) yi + 2 (a/Li — 61/ + c + p4^)^i
+ (1 + X» +/iA« + i;»a« 6»  c" 0«) Wi,
values which satisfy identically x^ V yf •\' zf •\ w^ ^ x^ •\ y^ '\ z^ •\ w^.
Dividing the linear equations by &, and forming with the coefficients on the right
hand side of the equation so obtained a determinant, the value of this determinant is
4 1 ; the transformation is consequently a proper one. And conversely, what is very
important, every proper transformation may be exhibited under the preceding form*.
I The nature of the reasoning by which this is to be established may be seen by considering the analogous
relation for two variables. Suppose that x^, y^ are linear functions of x and y such that x^{y^=a^+y'*\ then
if 2f=« + ari, 2iy=y+yi, ^, 17 will be linear functions of ar, y such that ?+i;*=$«+i;y, or $(f«) + i7(i7y) = 0;
^x must be divisible either by ri or else by 17y. On the former supposition, calling the quotient y, we have
:c=(Fi}, and thence ^=^(+17, leading to a transformation such as is considered in the text, and which is a proper
transformation; the latter supposition leads to an improper transformation. The given transformation, assumed
to be proper, exists and cannot be obtained from the second supposition; it must therefore be obtainable from
the first supposition, i.e. it is a transformation which may be exhibited under a form such as is considered in
the text.
122] A SURFACE OF THE SECOND ORDER INTO ITSBLF. 109
Next considering the equations connecting x, y, z, w with {, 97, (^, od, we see that
+ ( MfA'^+ ?+Cft))»
+ (af 617 cf+ ft))*.
We are thus led to the discussion (in connexion with the question of the trans
formation into itself of the form a" + y* + ^ + ti;*) of the new form
+ (vx+ y + 7iz + bvif
+ ( /jLx — \y+ z + cwy
+ (— oo? — 6y — c^4 w)";
or, as it may also be written,
(af + y* + j^{'tul') + (vyfiz{'awy{(\z — vx + bwy + (jix\y\'C^
Represent for a moment the forms in question by U, F, and consider the surfaces
[7=0, F=0. If we form from this the sur&ce V+qU=0, and consider the dis
cnminant of the function on the lefthand side, then putting for shortness
/c = V + /Lt« + i/« + a« + 6»+c>,
this discriminant is
which shows that the sur&ces intersect in four lines. Suppose the discriminant vanishes;
we have for the determination of 9 a quadratic equation, which may be written
5« + (2 + /e)5 + ir=0;
let the roots of this equation be q,, q,/, then each of the functions q,U\V, q,,U^ V
will break up into linear factors, and we may write
q,U+V^RA>
q,,U+r^RAr
(U and V are of course linear functions of R^, and R„8,^) forms which put in
evidence the fact of the two sur&ces intersecting in four lines.
The equations
a^ + a?, = 2f, yi+yt«2i;, Zi+Zi^2^, Wi + t£;, = 2fii,
no ON THE HOMOGRAPHIC TRANSFORMATION OF [122
show that the point (f, 17, f, «) lies in the line joining the points (a^i, yi, z^ Wi) and
(^11 3/21 ^21 wjj); aiid to show that this line touches the surface F=0, it is only
necessary to form the equation of the tangent plane at the point (f, 1;, (Ti <») o{ the
8ur£BU3e in question ; this is
(x + vy fjLZ + at(;) (f + V17 — ft(r + ck») + ... = ;
or what is the same thing,
{x •\ vy "' fiz •¥ aw) a?i + . . . = 0,
which is satisfied by writing (a?,, yi, i^i, Wj) for (x, y, e, w), that is, the tangent plane of
the surface contains the point (xi, y^ Zi, w,). We see, therefore, that the line through
(^, yii ^1. «^i) aiid (^> yjj 8^2 » ^2) touches the surface F=0 at the point (f, 17, f, a>).
Write now
^=~a'' ^=~r"' ^=":r» ^=^"a"» f^'^'iT' ^="^5
<p 9 9 9 9 9
if we derive from the coordinates a?!, yi, ^Ti, Wi, by means of these coefficients
a\ b\ c\ \'y f/, v\ new coordinates in the same way as «i, y„ z^ w^ were derived by
means of the coefficients a, 6, c, X, /li, r, the coordinates so obtained are — a?i, — yi,~z^i, — Wi,
i.e. we obtain the very same point (x^, y,, ^si ^i) by means of the coefficients (a, 6, c, X, /a, i;),
and by means of the coefficients (a\ h\ c\ \\ /li', v^). Call f, rf, f, oi' what f, 17, f, c»
become when the second system of coefficients is substituted for the first; the point
^j V> (r'» ^' will be a point on the sur&oe F' = 0, where
F'=<^>(a;» + y« + ^ + ti;»)
+ (— cy + 62: — Xw)* + (— 0^ i: ca? — /it^;)* + (— 6a: + ay — j/u;)" + (— Xr — /Lty — i/^)* ;
and since
F+ F' = /e(a» + y« + j^» + t£;»),
and F=0 intersects the surface iB* + y* + ^ + tt;* = in four lines, the surface F' =
will also intersect this surface in the same four lines. And it is, moreover, clear that
the line joining the points {x^, yi, Zi, Wi) and (x^, y,, ^„ n;,) touches the surfieu^ F' =
in the point (f', 17', f, co'). We thus arrive at the theorem, that when two points
of a surfisu^ of the second order are so connected that the coordinates of the one
point are linear functions of the coordinates of the other point, and the transformation
is a proper one, the line joining the two points touches two sur&ces of the second
order, each of them intersecting the given surface of the second order in the same
four lines. Any two points so connected may be said to be corresponding points, or
simply a pair. Suppose the four lines and also a single pair is given, it is not for
the determination of the other pairs necessary to resort to the two auxiliary surfaces
of the second order ; it is only necessary to consider each point of the surface as
determined by the two generating lines which pass through it; then considering first
122] A SURFACE OF THE SECOND ORDER INTO ITSELF. Ill
one point of the given pair, and the point the corresponding point to which has to
be determined, take through each of these points a generating line, and take also
two generating lines out of the given system of four lines, the four generating lines
in question being all of them of the same set, these four generating lines inter
secting either of the other two generating lines of the given sjrstem of four lines in
four pointa Imagine the same thing done with the other point of the given pair
and the required point, we should have another system of four points (two of them
of course identical with two of the points of the firstmentioned system of four points);
these two systems must have their anharmonic ratios the same, a condition which
enables the determination of the generating line in question through the required
point: the other generating line through the required point is of course determined
in the same manner, and thus the required point (i.e. the point corresponding to any
point of the surface taken at pleasure) is determined by means of the two generating
lines through such required point.
It is of course to be understood that the points of each pair belong to two
distinct systems, and that the point belonging to the one system is not to be con
founded or interchanged with the point belonging to the other system. Consider, now,
a point of the surface, and the line joining such point with its corresponding point,
but let the corresponding point itself be altogether dropped out of view. There are
two directions in which we may pass along the surface to a consecutive point, in
such mamier that the line belonging to the point in question may be intersected by
the line belonging to the consecutive point. We have thus upon the surface two
series of curves, such that a curve of each series passes through a point chosen at
pleasure on the surface. The lines belonging to the curves of the one series generate
a series of developables, the edges of regression of which lie on one of the surfaces
intersecting the surfeice of the second order in the four given lines; the lines belonging
to the curves of the other series generate a series of developables, the edges of
regression of which lie on the other of the surfaces intersecting the surface in the
four given lines; the general nature of the system may be understood by considering
the system of normals of a surface of the second order. Consider, now, the surface
of the second order as given, and also the two surfaces of the second order inter
secting it in the same four lines ; from any point of the sur&ce we may draw to
the auxiliary surfiEtces four dififerent tangents ; but selecting any one of these, and
considering the other point in which it intersects the surface as the point corre
sponding to the first*mentioned point, we may, as above, construct the entire system
of corresponding points, and then the line joining any two corresponding points will
be a tangent to the two auxiliary surfSaces; the system of tangents so obtained may
be called a system of congruent tangents. Now if we take upon the surface three
points such that the first and second are corresponding points, and that the second
and third are corresponding points, then it is obvious that the third and first are
corresponding points ;— observe that the two auxiliary surfaces for expressing the corre
spondence between the first and second point, those for the second and third point,
and those for the third and first point, meet the surface, the two auxiliary surfaces
of each pair in the same four lines, but that these systems of four lines are different
112 ON THB HOMOGRAPHIC TRANSFOBMATION OF A SURFACE &C. [122
for the different pairs of auxiliary surfaces. The same thing of course applies to any
numb^ of corresponding points. We have thus, finally, the theorem, if there be a
polygon of (m + 1) sides inscribed in a surfiEtce of the second order, and the first side
of the polygon constantly touches two surfaces of the second order, e8w;h of them
intersecting the surfisu^ of the second order in the same four lines (and the side
belong always to the same system of congruent tangents), and if the same property
exists with respect to the second, third, &c.... and wth side of the polygon, then will
the same property exist with respect to the (m + l)th side of the polygon.
We may add, that, instead of satisfying the conditions of the theorem, any two
consecutive sides of the polygon, or the sides forming any number of pairs of con
secutive sides, may pass each through a fixed point. This is of course only a
particular case of the improper transformation of a surfietce of a second order into
itself, a question which is not discussed in the present paper.
1231
113
123.
ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN
INTEGRAL.
[From the Philosophical Magazine, vol. vi. (1853), pp. 414 — 418.]
The equation of a surface passing through the curve of intersection of the surfaces
a^+ y*+ ^'+ u;* = 0,
flw^ + 6y* + cz^ H dv)* = 0,
is of the form
»(a^ + y' + 2' + t£;') + aa^ + 6y' + c^ + dti;* = 0,
where tt is an arbitrary parameter. Suppose that the sur&ce touches a given plane,
we have for the determination of 8 a cubic equation the roots of which may be
considered as parameters defining the plane in question. Let one of the values of 8
be considered equal to a given quantity k, the plane touches the surface
aud the other two values of 8 may be considered as parameters defining the particular
tangent plane, or what is the same thing, determining its point of contact with the
Burface.
Or more clearly, thus: — in order to determine the position of a point on the
8ur£BM^e
A; (a;* + y2 + ^* + w") + aa?» + 6y« + C2:« + dt(;» = ;
the tangent plane at the point in question is touched by two other surfaces
/> (^ + y ' + ^ + 1^;") + cue" + 6y * + c^* + dw" = 0,
9(«' + y* + '8^* + w*) + cuc» + 6y' + C2:* + dt£;» = 0;
c. n. 15
114 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [l23
and, this being so, p and q are the parameters by which the point in question is
determined. We may for shortness speak of the surface
A: (iB» + y» + ^ + w") + oa^ + 6^ + ce' + dw" =
as the surfiiK^ (jk). It is clear that we shall then have to speak of
ai^ + y^ + z^ + v)^ =
as the surface (oo).
I consider now a chord of the surface (oo) touching the two surfaces (k) and
(k')] and I take 0, <l> aa the parameters of the one extremity of this chord; (p, q)
as the parameters of the point of contact with the surface (A;); p', q' as the parameters
of the point of contact with the surface (kf); and O', ^' as the parameters of the
other extremity of the chord; the points in question may therefore be distinguished
as the points (oo ; ^, 0), (i; p, g), (k' \ p\ g'), and (oo; ^, ^'). The coordinates of the
point (oo ; 0, <f>) are given by
X : y : z : w= V(a + ^) (a + 0) : V(a — 6) (a — c) (a  d)
V(6+^)(6 + </>) r V(6c)(6d)(6a)
V(c + ^) (c + 4>) r V(c  d) (c  a) (c  6)
^/(dTW(d + 'f) H V(da)(d6)(dc) ;
those of the point (k; p, q) by
X ', y : z : w^ V(a +p) (a + j) ^ V(a — 6) (a — c) (a — d) Va + i
V(6+p)(6 + 5) ^ V'(6c)(6d)(6a) ViTib
V(c + p) (c + g) r V(c  d) (c  a) (c  6) Vc+i
V(d+/>)(d + g)f.V(da)(d6)(dc) Vd+l;
and similarly for the other two points.
Consider, in the first place, the chord in question as a tangent to the two
surfaces (£) and {kf). It is clear that the tangent plane to the surface {k) at the
point {k\ p, q) must contain the point (Ar'; p\ <f)^ and vice versd. Take for a moment
f, i;, (r> ® ®* *^® coordinates of the point (A; p, g), the equation of the tangent
plane to (£) at this point is
2(a + A:)fd: = 0;
or substituting for f,... their values
S (a;V(a+p)(a + g) Va {Ic r V(a  6) (a  c) (a  d) ) = ;
123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELTAN INTEGRAL. 115
or taking for a?,... the coordinates of the point (kf^ p\ ^\ we have for the conditions
that this point may lie in the taDgent plane in question,
or under a somewhat more convenient form we have
2((6c)(cd)(d6)V(a+p)(a + ?)V(a + p')(a4gO^^) = 0,
for the condition in order that the point (Je\ p\ q') may lie in the tangent plane at
(^> P* ?) ^ *^® surface {k). Similarly, we have
2f(6c)(ccO(d6)V(a+i>)(a + g)V(a+/)0(a + 9')^5^')=O,
\ ya\ kJ
for the condition in order that the point (A?, p, q) may lie in the tangent plane at
ifff\ p\ 5O ^ *^® surfiwje Qf). The former of these two equations is equivalent to
the sjrstem of equations
V(a +/>) (a + q) (a +;>0 (« + 9') V l^A/ = ^ + A^ + ^'>
and the latter to the system of equations
V(o +p) (o + q) (a + p') (a + q) y ^ = V + /t'o + v'a*;
where in each system a is to be successively replaced by 6, c, d, and where X, /ia, 1/
and V, ;i', 1/ are indeterminate. Now dividing each equation of the one system by
the corresponding equation in the other system, we see that the equation
x\k _ \ { fix + va^
^ satisfied by the values a, h, c, d oi x\ and, therefore, since the equation in a? is
^^y of the third order, that the equation in question must be identicaUy true. We
naay therefore write
\ + fix\va^=^(px^<T)(x + k\ \' + fix+v*a^ = (px + a)(x\'^),
*^^ the two systems of equations become therefore equivalent to the single system,
V(a +p) (a + q) (a +/) (a f q') = (pa + a) V(a + *)(« + *'),
V(6+;?)(6 + 9)(6+;?')(6 + ?') = (p6 + <^) V(ft + A) (6 + A:'),
V(c +p) (c + q) (c + Jt>0 (c + 9') = (pc + cr) V(c + A) (c + A:'),
V(d +i>) (d + ?) (d +p') {d h gO = (pd + cr) V(d + A:) (d + jfc'),
15—2
116 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [123
a set of equations which may be represented by the single equation
where x is arbitrary; or what is the same thing, writing —a: instead of a?,
Hence, putting
J v(a? + a)(a: + 6)(a: + c)(a? + cO(a?A:)(a?A?')*
J V(a?4a)(a? + 6)(a? + c)(a? + c0(a?A:)(a?ifc')
we see that the algebraical equations between p, q\ p\ <]( are equivalent to the
transcendental equations
Up ±liq ± Up' ± Ilgr' = const.
TI,p ± U^q ± ny ±U,q' = const.
The algebraical equations which connect 0, <f> with p, q; p\ ((, may be exhibited
under several different forms; thus, for instance, considering the point (oo ; ^, </>) as
a point in the line joining (A; p, q) and (A:'; p\ q% we must have
V(a+p)(a + g) ^ VoTA, V(6 4p)(6 + g') ^ \/6 + *,. . .
= 0,
i.e. the determinants formed by selecting any three of the four columns must vanish;
the equations so obtained are equivalent (as they should be) to two independent
equations.
Or, again, by considering (oo ; d, ^) first as a point in the tangent plane at
{k\ p, q) to the surface (k), and then as a point in the tangent plane at (Ar^; p\ q')
to the surface (&'), we obtain
2 ((6 c)(cd)(d  6) V(a + p)(a + g) V(aTl) V(a + ^)(a4<^)) = 0,
2((6c)(c(0(d6)V(^TpOMY^
Or, again, we may consider the line joining (oo ; 6, <f>) and (k; p, q) or (A/; p\ q'),
as touching the surfaces (k) and (A/); the formulae for this purpose are readily
obtained by means of the lemma, —
123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. 117
" The condition in order that the line joining the points (f , i;, (^, ©) and (f', n\ ^, (o')
inay touch the surface
is
2ab(fi7'r^)^ = 0.
the summation extending to the binary combinations of a, b, c, dr
But none of all these formulae appear readily to conduct to the transcendental
equations connecting 0, with p, q; p\ q\ Reasoning from analogy, it would seem
that there exist transcendental equations
±U0 ±U<f} tllp ± Up' =con8t.
± n^0 ± n,<f> ± U^p ± U^p' = const.,
or the similar equations containing q, ((, instead of j}, p\ into which these are changed
by means of the transcendental equations between j), q, p\ ((. If in these equations
we write ff^ (f/ instead of 0, ^, it would appear that the functions Up, lip', II^p, Tl^p'
may be eliminated, and that we should obtain equations such as
±T10 ±U(I> ±U0' ± n<f)' = const.
± U,0 ± U^<f> ± n,^ ± n,</)' = const.
to express the relations that must exist between the parameters 0, <(> and ^, <!>' of
the extremities of a chord of the surface
ic» + y» + ^ + t(;* = 0,
b order that this chord may touch the two surfaces
* (^ H y* + ^ + 1^;") + our* H 6y* + c^ + dt£;» = 0,
Ar'(aj» + y» + ^ + «(;») f aa;* H 6y* H c^ + dw" = 0.
The quantities k, V, it will be noticed, enter into the radical of the integrals
JIo?, II/p. This is a very striking difiference between the present theory and the
analogous theory relating to conies, and leads, I think, to the inference that the theory
of the polygon inscribed in a conic, amd the sides of which tovdt conies intersecting
the conic in the same four points, cannot be extended to surfaces in such manner as
one might be led to suppose from the extension to surfaces of the much simpler
theory of the polygon inscribed in a conic, and the sides of which totich conies having
double conta^ct with the conic, (See my paper "On the Homographic Transformation
of a 8ur£Eu;e of the second order into itself," [122]).
The preceding investigations are obviously very incomplete; but the connexion
which they point out between the geometrical question and the Abelian integral
involving the root of a function of the sixth order, may I think be of service in
the theory of these integrals.
118
[124
124
ON A PEOPEETY OF THE CAUSTIC BY EEFEACTION OF THE
CIECLE.
[From the Philosophical MagcLzine, vol vi. (1853), pp. 427 — 431.]
M. St Laurent has shown (Oergonne, vol. xviu. [1827] p. 1), that in certain cases
the caustic by refiraction of a circle is identical with the caustic of reflexion of a circle
(the reflecting circle and radiant point being, of course, properly chosen), and a very
elegant demonstration of M. St Laurent's theorems is given by M. Gergonne in the
same volume, p. 48. A similar method may be employed to demonstrate the more
general theorem, that the same caustic by refraction of a circle may be considered as
arising from six different systems of a radiant point, circle, and index of refr'action.
The demonstration is obtained by means of the secondary caustic, which is (as is well
known) an oval of Descartes. Such oval has three foci, any one of which may be
taken for the radiant point: whichever be selected, there can always be found two
corresponding circles and indices of refraction. The demonstration is as follows : —
Let c be the radius of the refiucting circle, /a the index of refraction; and taking
the centre of the circle as origin, let f, 17 be the coordinates of the radiant point,
the secondary caustic is the envelope of the circle
where a, /3 are parameters which vary subject to the condition
the equation of the variable circle may be written
{^2(a^ + y« + c«)(f> + i7« + c')}2(/i«a:f)a2(/i»yi7)/9 = 0,
124] ON A PROPERTY OP THE CAUSTIC BY REFRACTION OP THE CIRCLE. 119
which is of the form
the envelope is therefore
Hence substituting, we have for the equation of the envelope, Le. for the secondary
caustic,
(A^^a^ + y' + c»)  (P + 17' + c»)}» = 4c» {(Ai'a;  f )« + (M»y  17)»},
which may also be written
and this may perhaps be considered as the standard form.
To show that this equation belongs to a Descartes' oval, suppose for greater con
venience 17 = 0, and write
/Lt*(a5» + y" c*)  p + c* = 2c/i V(a?f)» + y» ;
1 . / IV
multiplying this equation by 1 — 5, and adding to each side c^lfi ) 4.(3? — f)* + y«,
we have
(i^.){M»(a^+y'c)?+c}+(«f)'+y'+c(A*iy
= (a;f)» + y» + 2c(/t*i)V(xf)» + y' + c»(My*;
0' Kdacing
Bgun, multiplying the same equation by — (l — 5), and adding to each side
we have
^' reducing,
(«_^' + y. = V(a,_f). + y. + (lDp.
120 ON A PROPBRTY OP THE CAUSTIC BY REFRACTION OF THE CIRCLE, [124
Hence, extracting the square roots of each side of the equations thus found, we
have the equation of the secondary caustic in either of the forms
to which are to be joined
Vi'^'
^^=f\/(
.'M'j ^y
c(.y7(.g.^.(
f.^y(.
^hy''
/fr fA.^\
.(_^)V(«;f). + y.
Write successively,
r=f .
c' = c ,
/t*' = M ,
(1)
A*
/ c
(«)
^■'\
(/9)
r=f .
"'? •
(7)
c' = Cf
"T'
(«)
(0
or, what is the same thing.
f=r .
c = c',
^=/*' .
(1)
(«)
c=,
1
(/3)
f=r,
r
(7)
CSC*,
(8)
V 1:' >
(Z
''=/•
c'
(«)
r=f .
H'
 =?
A*" ^ '
H
r=f .
124] ON A PROPERTY OF THE CAUSTIC BY REFRACTION OF THE CIRCLE. 121
or, again,
(1)
(«)
(7)
(8)
(*)
then, whichever system of values of f, c', /Lt' be substituted for f, c, /a, we have in
each case identically the same secondary caustic, the effect of the substitution being
amply to interchange the different forms of the equation; and we have therefore
identically the same caustic. By writing
&c.,
^ j9, 7, 5, € will be functional symbols, such as are treated of in my paper " On the
Theory of Groups as depending on the symbolic equation 6^ = 1," [126], and it is
^ to verify the equations
= ^8" = 87 = eS = 7€,
/9s= a? = ey = 78 = Se,
7 = Sa = 6/9 = /98 = oe,
S = ea = 7/8 = 07 = /Se,
e = 7a = 8/9=/97 = aS.
Suppose, for example, f= — c, i.e. let the radiant point be in the circumference;
™n in the fourth system ^ = — c, c' = — , (or, since d is the radius of a circle, this
radius may be taken  ), /Lt' = — 1, or the new system is a reflecting sjrstem. This is
^^^ of M. St Laurent's theorems, viz.
C. IL 16
122 ON A PROPBRTY 0*" THE CAUSTIC BY REFRACTION OF THE CIRCLE. [124
Theorem. The caustic by refiractiou of a circle when the radiant point is on the
circumference, is the caustic by reflexion for the same radiant point, and a concentric
circle the radius of which is the radids of the first circle divided by the index of
re&action.
Again, if f = — c/li, the fifth system gives ^ = ^^ , d — Cy /a' =^ — 1, or the new system
is in this case also a reflecting system. This is the other of M. St Laurent's
theorems, viz. : —
Theorem. The caustic by refraction of a circle when the distance of the radiant
point bom the centre is equal to the radius of the circle multiplied by the index of
refiraction, is the caustic by reflexion of the same circle for a radiant point which Ls
the image of the first radiant point.
Of course it is to be understood that the image of a point means a point whose
distance firom the centre = square of radius r distance.
2 Stom Buildings, Nov. 2, 1863.
125]
123
125.
ON THE THEORY OF GROUPS, AS DEPENDING ON THE
SYMBOLIC EQUATION ^=1.
[From the Philosophical Magazine, vol. vii, (1854), pp. 40 — 47.]
Let ^ be a symbol of operation, which may, if we please, have for its operand,
not a single quantity x, but a sjrstem (x, y, ...)> ^^ ^^^^
0(^, y, ...) = («^. y', ...),
where a/, y', ... are any functions whatever of x, y, ..., it is not even necessary that
x\ y\ ... should be the same in number with x, y, ..,. In particular a/, y', &c. may
represent a permutation of x, y, &c., is in this case what is termed a substitution;
and if^ instead of a set x, y, ..., the operand is a single quantity x, so that Ox^af =fx,
fl is an ordinary functional symbol. It is not necessary (even if this could be done)
to attach any meaning to a symbol such as tf ± <^, or to the symbol 0, nor con
sequently to an equation such as ^ = 0, or ^±^ = 0; but the symbol 1 will naturally
denote an operation which (either generally or in regard to the particular operand)
leaves the operand unaltered, and the equation = <l> will denote that the operation
is (either generally or in regard to the particular operand) equivalent to kJ), and
of course 0=1 will in like manner denote the equivalence of the operation to the
operation 1. A sjnmbol 0<l> denotes the compound operation, the performance of which
is equivalent to the performance, first of the operation (f>, and then of the operation
0; 0^ is of course in general different from ^^. But the symbols 0, 0, ... are in
general such that ^ . ^ = ^0 . %> &c., so that 0^, 0<lyx^t ^^ ^^^^ ^ definite signi
fication independent of the particular mode of compounding the symbols; this will
be the case even if the functional operations involved in the sjmcibols 0, ^, &c.
contain pajrameters such as the quaternion imaginaries i, j, k; but not if these
iiinctional operations contain parameters such as the imaginaries which enter into the
theory of octaves, &c., and for which, e.g. a.l3y is something different from a^.7,
16—2
124
ON THE THEORY OF GROUPS,
[125
a supposition which is altogether excluded from the present paper. The order of the
factors of a product ^^;^... must of course be attended to, since even in the case
of a product of two factors the order is material; it is very convenient to speak of
the symbols 6, ^ . . . as the first or furthest, second, third, &c., and last or nearest
factor. What precedes may be almost entirely summed up in the remark, that the
distributive law has no application to the symbols ^^ ... ; and that these symbols are
not in general convertible, but are associative. It is easy to see that ^=1, and
that the index law ^.^=6?^+'*, holds for all positive or negative integer values,
not excluding 0. It should be noticed also, that if tf = 0, then, whatever the symbols
o, /8 may be, a0ff = a<f>0, and conversely.
A set of symbols,
1, a, ^, ...
all of them different, and such that the product of any two of them (no matter in
what order), or the product of any one of them into itself, belongs to the set, is
said to be a groups. It follows that if the entire group is multiplied by any one
of the symbols, either as further or nearer factor, the effect is simply to reproduce
the group; or what is the same thing, that if the symbols of the group are multi
plied together so as to form a table, thus:
Farther factors
1 a /3 ..
to
a
13
1
a
a
)8«
a*
/9
•
•
«/3
/S*
that as well each line as each column of the square will contain all the symbols
1, a, ^, — It also follows that the product of any number of the S3mibols, with or
Avithout repetitions, and in any order whatever, is a symbol of the group. Suppost^
that the group
1, a, ^, ...
contains n symbols, it may be shown that each of these sjrmbols satisfies the equation
^ = 1;
so that a group may be considered as representing a system of roots of this symbolic
binomial equation. It is, moreover, easy to show that if any symbol o of the group
1 The idea of a group as applied to permatations or substitutions is due to Oalois, and the introduction
of it may be considered as marking an epoch in the progress of the theory of algebraical equations.
125]
AS DEPENDING ON THE SYMBOLIC EQUATION ^=1.
125
satisfies the equation ^ = 1, where r is less that n, then that r must be a sub
multiple of n; it follows that when n is a prime number, the group is of necessity
of the form
1, a, a\...a^\ (a«=l);
and the same may be (but is not necessarily) the case, when n is a composite
number. But whether n be prime or composite, the group, assumed to be of the
form in question, is in every . respect analogous to the system of the roots of the
ordinary binomial equation a:* — 1 = ; thus, when n is prime, all the roots (except
the root 1) are prime roots; but when n is composite, there are only as many prime
roots as there are numbers less than n and prime to it, &c.
The distinction between the theory of the symbolic equation ^ = 1, and that of
the ordinary equation a::* — 1 = 0, presents itself in the very simplest case, n = 4. For,
consider the group
which are a system of roots of the symbolic equation
There is, it is clear, at least one root /3, such that /S" = 1 ; we may therefore
represent the group thus,
1, a, /9, al3, (i8» = l);
then multiplying each term by a as further factor, we have for the group 1, o*, afi,
a'jS, so that a' must be equal either to or else to 1. In the former case the
group is
1, a, a\ a», (a*=l),
which is analogous to the system of roots of the ordinary equation ic* — 1 = 0. For
the sake of comparison with what follows, I remark, that, representing the last
mentioned group by
1, a, A 7»
we have the table
1,
a,
A
13
1
a
13
7
7
a
13
13
1
7
1
a
/3
7
1
a
126
ON THE THEOllY OF GEOUP8,
[125
If, on the other hand, ct* = l, then it is easy by similar reasoning to show that we
must have a^ = /3cc, so that the group in the case is
1, a, /9, a^e, (a« = l, /3» = 1, a/8 = /3a);
or .if we represent the group by
we have the table
1 a i8 7
/9
1
a
/9
7
a
1
7
7
1
a
i
1
7
/9
a
or, if we please, the symbols are such that
a« = /3» = 7» = 1,
/9 = 7« = «A
7 =a/9 = i8a;
[and we have thus a group essentially distinct from that of the system of roots of
the ordinary equation ic* — 1 = 0].
Systems of this form are of frequent occurrence in analysis, and it is only on
account of their extreme simplicity that they have not been expressly remarked. For
instance, in the theory of elliptic functions, if n be the parameter, and
/ \ ^ Q/ \ ^ + ^ / \ c»(l+n)
then a, /8, y form a group of the species in question. So in the theory of quadratic
forms, if
a (a, 6, c) = (c, 6, a)
0(a, 6, c) = (a, 6, c)
7 (a, 6, c) = (c,  6, a) ;
although, indeed, in this case (treating forms which are properly equivalent as identical)
we have a = /8, and therefore 7=1, in which point of view the group is simply a
group of two symbols 1, «,(«'=!).
125] AS DEPENDING ON THE SYMBOLIC EQUATION ^ = 1. 127
Again, in the theory of matrices, if I denote the operation of inversion, and tr
that of transposition, (I do not stop to explain the terms as the example may be
passed over), we may write
a = 7, i8 = tr, 7 = 7 . tr = tr . 7.
I proceed to the case of a group of six symbols,
1, a, )8, 7, 8, e,
which may be considered as representing a system of roots of the symbolic equation
It is in the first place to be shown that there is at least one root which is a
prime root of ^ = 1, or (to use a simpler expression) a root having the index 3. It
is clear that if there were a prime root, or root having the index 6, the square of
this root would have the index 3, it is therefore only necessary to show that it is
impossible that all the roots should have the index 2. This may be done by means
of a theorem which I shall for the present assume, viz. that if among the roots of
the symbolic equation ^ = 1, there are contained a system of roots of the symbolic
equation 6^=1 (or, in other words, if among the symbols forming a group of the order
there are contained symbols forming a group of the order j>), then j> is a submultiple
of n. In the particular case in question, a group of the order 4 cannot form part
of the group of the order 6. Suppose, then, that 7, h are two roots of ^=1, having
each of them the index 2; then if 7S had also the index 2, we should have 7S = S7;
and 1, 7, 8, S7, which is part of the group of the order 6, would be a group of
the order 4. It is easy to see that 7S must have the index 3, and that the group
is, in fact, 1, 78, S7, 7, 8, 7S7, which is, in fact, one of the groups to be presently
obtained; I prefer commencing with the assumption of a root having the index 8.
Suppose that a is such a root, the group must clearly be of the form
1, o, o*, 7, a7, a«7, (a»=l);
and multipljring the entire group by 7 as nearer factor, it becomes 7, 07, a*7, 7,
ay, fltV; we must therefore have 7^ = 1, a, or a". But the supposition 7* = a' gives
y* = a* = a, and the group is in this case 1, 7, 7*, 7*, 7*, 7" (7^=1); and the suppo
sition 7* = a gives also this same group. It only remains, therefore, to assume 7* = 1 ;
ihen we must have either 7a = a7 or else 70 = 0*7. The former assumption leads to
the group
1, a, a", 7, 07, a^ (a» = l, 7*= I, l^^^l\
which is, in fact, analogous to the system of roots of the ordinary equation afi—\ = ^\
and by putting 07 = X, might be exhibited in the form 1, \, V, X', \*, V, (\« = 1),
under which this system has previously been considered. The latter assumption leads
to the group
1, o, a«, 7, 07, o^ (a» = l, 7^ = 1, 7a = «^)>
and we have thus two, and only two, essentially distinct forms of a group of six.
If we represent the first of these two forms, viz. the group
1, «, a*, 7» «% a'Vi (a' = l, 7' = 1> 7a=«7)
128
ON THE THEORY OF OBOUPS,
[125
by the general symbols
we have the table
1. a. /9, 7, S, e,
1. a. /9,
S,
1
a
1
7
S
t*
a
fi
7
8
e
1
/9
7
S
e
1
a
7
B
e
1
a
/9
ys
b
€
1
a
7
€
1
a
P
7
S
while if we represent the second of these two forms, viz. the group
1, a, a\ 7, 07, o^ (ct* = l, 7»=1, 70 = 0*7),
by the same general symbols
1, a, /9, 7» S» €,
we have the table
10/9786
1
/8
1
a
ys
7
S
e
a
/9
1
e
7
S
y3
1
a
S
€
7
7
£
e
1
iS
S
€
7
^
1
a
e
7
h
1
125] AS DEPENDING ON THE SYMBOLIC EQUATION $*'=l. 129
or, what is the same thing, the system of equations is
1 = ^a = 5r/9 = 7» =55=6*,
a = /8^ = Sy = eS = ye,
/8 = a' = €7 = yS = Se,
fy = Set = e^ = ^g = a€,
S = ea = y/8 = ay = /8e,
£ = fya = S/8 = /9y = aS.
An instance of a group of this kind is given by the permutation of three letters;
the group
1, a, 0f % ^» f
may represent a group of substitutions as follows : —
abc, cahy bca, acb, cba, bac
abc abc abc abc abc abc.
Another singular instance is given by the optical theorem proved in my paper
"On a property of the Caustic by refraction of a Circle, [124]."
It is, I think, worth noticing, that if, instead of considering a, /3, &c. as symbols
of operation, we consider them as quantities (or, to use a more abstract term, *cogi
tables') such as the quaternion imaginaries; the equations expressing the existence
of the group are, in fact, the equations defining the meaning of the product of two
complex quantities of the form
w + aa + 6/8 4 . . . ;
thus, in the system just considered,
(wHaa + 6/9 + cy + dS + C6)(w;' + a'a + 6'/8 + cy + d'S + e'6)= W + Aa + B/S + Cy + DS + Ee,
where
W = tuw' + a6' + a'6 + cc' + dd' H ee\
A =wa' + iji/a+ bV + dc' + edf + cc',
5 = w6' + w/6 4 aa'\ ed + cd' + de\
C =^wc' + w/c 4 da' + eV + 6d' + aefy
D = wd! +v/d{ea' + cb' + ac' + be\
E =we' + w'e H ca' + d6' + 6c' ^adf.
It does not appear that there is in this system anjrthing analogous to the
Baodulus <i;* + a^ + y* + ?', so important in the theory of quatemiona
I hope shortly to resume the subject of the present paper, which is closely
connected, not only with the theory of algebraical equations, but also with that of
c. n. 17
130 ON THE THEORY OF GROUPS, &C. [125
the composition of quadratic forms, and the 'irregularity' in certain cases of the
determinants of these forms. But I conclude for the present with the following two
examples of groups of higher orders. The first of these is a group of eighteen, viz.
1, a, A 7, ayS, i8a, 07, 70, /97, 7/8, a/87, /97a, 70/8, a^a, ^87^ 70:7, a^rtP, ^7^a,
where
ct« = l, /9» = 1, 7» = 1, (/87)» = 1, (7«)' = 1, («^)' = 1, (a/87)» = l, (/37a)» = l, (7«/8)» = l;
and the other a group of twentyseven, viz.
1, a, Qi*, 7» y* 7«, «% 7«'» aY t"** ^T^i 7^. «V»
07a, 07*a, 0*70:, 0*7*0, 070*, 07*0*, o^ct*, o^a", 707', 70*7*, 7*07, 7*0*7, 7*070*, 707*0*,
where
o» = l, 7»=1, (7o)' = l, (yo)» = l, (7Qt«)' = l, (7'c^)'=l.
It is hardly necessary to remark, that each of these groups is in reality perfectly
symmetric, the omitted terms being, in virtue of the equations defining the nature
of the symbols, identical with some of the terms of the group: thus, in the group
of 18, the equations a? = l, /9* = 1, 7^ = 1, (0/87)* = 1 give 0^7 = 7/80, and similarly for
all the other omitted term& It is easy to see that in the group of 18 the index
of each term is 2 or else 3, while in the group of 27 the index of each term is 3.
2 SUyns Buildings, Nov. 2, 1853.
126]
131
126.
ON THE THEOKY OF GKOUPS, AS DEPENDING ON THE
SYMBOLIC EQUATION <9»=1.— Second Part.
[From the PkUoaophical Magazine, vol. vii. (1854), pp. 408 — 409.]
Imagine the symboU
L, M, N, ...
Bach that (JL being any symbol of the system),
w the group
1, a, A;
then, in the first place, M being any other 83rmbol of the system, M^^L, M^^M,
^''%,.. will be the same group 1, a, /9, In fact, the system Z, M, N,.., may be
^tten Z, Za, L8...; and if e.g. if = Za, iV'=Z/9 then
M^N = (La)' Z/9 = a* Z"^ Z/9 = a%
which belongs to the group 1, a, A ....
Next it may be shown that
LL'\ ML\ NL\...
^ & group, although not in general the same group as 1, a, ^, .... In bet, writing
^^ia, N=Lfi, &c., the i^mbols just written down are
LL''\ LolL\ Z/9ZS...
*^d we have e.g. LoJr' . L/SL"' = LajSL"' = LyL''\ where 7 belongs to the group I, a, 0,
17—2
132 ON THE THEORY OF GROUPS, &C. [126
The system Z, M, JV, ... may be termed a groupholding system, or simply a
holder; and with reference to the two groups to which it gives rise, may be said
to hold on the nearer side the group Z~*Z, L~^M, Z""W, ..., and to hold on the
further side the group ZZ~S LM"^, LN~^,.., Suppose that these groups are one and
the same group 1, a, 13... y the system Z, Jf, JT, ... is in this case termed a sym
metrical holder, and in reference to the lastmentioned group is said to hold such
group symmetrically. It is evident that the symmetrical holder Z, M, N, ... may be
expressed indiflferently and at pleasure in either of the two forms Z, Zo, Lj3,... and
Z, ctZ, I3L', Le. we may say that the group is convertible with any symbol Z of
the holder, and that the group operating upon, or operated upon by, a symbol Z of
the holder, produces the holder. We may also say that the holder operated upon by,
or operating upon, a symbol a of the group reproduces the holder.
Suppose now that the group
li a, A y> 8, €, f. •••
can be divided into a series of symmetrical holders of the smaller group
1, a, ^, ... ;
the former group is said to be a multiple of the latter group, and the latter group
to be a submultiple of the former group. Thus considering the two different forms
of a group of six, and first the form
1, a, «•, 7» 7«» 7a'> (a* = l, 7* = !. «7=7a),
the group of six is a multiple of the group of three, 1, a, o* (in fiwt, 1, a, o*
and 7, 70, 7a* are each of them a symmetrical holder of the group 1, a, a'); and
so in like manner the group of six is a multiple of the group of two, 1, 7 (in fact,
1, 7 and a, a7, and a, 0*7 are each a symmetrical holder of the group 1, 7). There
would not, in a case such as the one in question, be any harm in speaking of the
group of six as the product of the two groups 1, a, a' and 1, 7, but upon the whole
it is, I think, better to dispense with the expression.
Considering, secondly, the other form of a group of six, viz.
1, a, <^f 7» 7a, 7a"(a* = l, 7' = 1> a7 = 7a');
here the group of six is a multiple of the group of three, 1, a, a* (in fact, as be
fore, 1, a, a' and 7, 70, 70', are each a symmetrical holder of the group 1, o, o',
since, as regards 7, 7a, 7a", we have (7, 70, 70') = 7(1, o, o«) = (l, o", 0)7). But
the group of six is not a multiple of any group of two whatever; in feet, besides
the group 1, 7 itself, there is not any symmetrical holder of this group 1, 7; and
so, in like manner, with respect to the other groups of two, 1, 7a, and 1, 7a'. The
group of three, 1, a, a\ is therefore, in the present case, the only submultiple of
the group of six.
It may be remarked, that if there be any number of symmetrical holdera of the
same group, 1, a, )8, ... then any one of these holders bears to the aggregate of the
holders a relation such as the submultiple of a group bears to such group; it is
proper to notice that the aggregate of the holders is not of necessity itself a holder.
127]
133
127.
ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE
OF THE SECOND ORDER INTO ITSELF.
[From the Philosophical Magazine, vol. vil (1854), pp. 208 — 212: continuation of 122.]
I PASS to the improper transformation. Sir W. K Hamilton has given (in the note,
p 723 of his Lectures on Quaternions [Dublin, 1853)] the following theorem : — If there
l>e a polygon of 2m sides inscribed in a surface of the second order, and (2m  1) of
the sides pass through given points, then will the 2mth side constantly touch two
cones circumscribed about the surface of the second order. The relation between the
extremities of the 2mth side is that of two points connected by the general improper
^fansformation ; in other words, if there be on a surface of the second order two
points such that the line joining them touches two cones circumscribed about the
stir&ce of the second order, then the two points are as regards the transformation
^ question a pair of corresponding points, or simply a pair. But the relation between
^he two points of a pair may be expressed in a different and much more simple
form. For greater clearness call the surface of the second order Z7, and the sections
along which it is touched by the two cones, 0, <(>; the cones themselves may, it is
<^W, be spoken of as the cones 0, <f>. And let the two points be P, Q. The line
•PQ touches the two cones, it is therefore the line of intersection of the tangent
plane through P to the cone 0, and the tangent plane through P to the cone 0.
l^t one of the generating lines through P meet the section in the point A, and
the other of the generating lines through P meet the section ^ in the point B.
The tangent planes through P to the cones 0, if> respectively are nothing else than
the tangent planes to the surface U at the points A, B respectively. We have there
fore at these points two generating lines meeting in the point P; the other two
134 ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE [127
generating lines at the points A, B meet in like manner in the point Q. Thus P,
Q are opposite angles of a skew quadrangle formed by four generating lines (or, what
is the same thing, Ijdng upon the surface of the second order), and having its other
two angles, one of them on the section and the other on the section ^ ; and if
we consider the side PA as belonging determinately to one or the other of the two
systems of generating lines, then when P is given, the corresponding point Q is, it
is clear, completely determined. What precedes may be recapitulated in the statement,
that in the improper transformation of a surface of the second order into itself, we
have, as corresponding points, the opposite angles of a skew quadrangle lying upon
the surface, and having the other two opposite angles upon given plane sections of
the surface. I may add, that attending only to the sections through the points of
intersection of 0, ^, if the point P be situate anywhere in one of these sections,
the point Q will be always situate in the other of these sections, Le. the sections
correspond to each other in pairs; in particular, the sections 0, <f> are corresponding
sections, so also are the sections 6, <I> (each of them two generating lines) made by
tangent planes of the surface. Any three pairs of sections form an involution ; the
two sections which are the sibiconjugates of the involution are of course such, that,
if the point P be situate in either of these sections, the corresponding point Q will
be situate in the same section. It may be noticed that when the two sections 0, <f>
coincide, the line joining the corresponding points passes through a fixed point, viz.
the pole of the plane of the coincident sections; in fact the lines PQ and AB are
in every case reciprocal polars, and in the present case the line AB lies in a fixed
plane, viz. the plane of the coincident sections, the line PQ passes therefore through
the pole of this plane. This agrees with the remarks made in the first part of the
present paper.
The analytical investigation in the case where the sur&ce of the second order
is represented under the form xy — zw = is so simple, that it is, I think, w^orth
while to reproduce it here, although for several reasons I prefer exhibiting the final
result in relation to the form ic" + y* + ;?* + t(;* = of the equation of the surface of
the second order. I consider then the surface xyzw = 0, and I take (a, /9, 7, S),
(a\ /S', y, S') for the coordinates of the poles of the two sections 0, ^, and also
(^i> yi> ^i> ^i)> (^> ya» ^a» ^«) fts the coordinates of the points P, Q. We have of course
a^i^i — j^iWi = 0, flPi^a — jSjWj = 0. The generating lines through P are obtained by com
bining the equation xy — zw = of the surface with the equation wi/i + yxi^zwi^wzi^O
of the tangent plane at P. Eliminating a firom these equations, and replacing in the
result Xi by its value ^^, we have the equation
(y^  ^yi) (ywi  t(;yi) = 0.
We may if we please take yzi — zyi^O, asy^^yxi zWi^wzi^O as the equations of
the line PA ; this leads to
yzizyi^O, I yw;«w;y« = 0, 
«y 1 + y ^ "" 8^1 "" ^^1 = , j wy^ + y^a — ^«^j — ^^^ = o, j
^a  zy^ = 0, I
h + y^a — ^^a — t^^a = 0, j
127] OF THE SECOND ORDER INTO ITSELF. 135
for the equations of the lines FA^ QA respectively; and we have therefore the
coordinates of. the point A^ coordinates which must satisfy the equation
of the plane 0. This gives rise to the equation
ya (ayi  S^i)  J^a (7^1  )9^i) = 0
We have in like manner
ayi¥yx^''ZWiwzj = 0,] xy^
for the equations of the lines P£, QB respectively ; and we may thence find the
coordinates of the point 5, coordinates which must satisfy the equation
^x + a'y  h'z  7't(; =
of the plane ^. This gives rise to the equation
y% («Vi  7^1)  z^ (8^1  /S'wi)
It is easy, by means of these two equations and the equation x^^ — z^w^^O, to form
the system
a?a = (ayi  S^i) (a'yi  7^i)>
ya == (7yi  fiz^) i^'yi  ^^i)»
w/a = (ayi  S?,) (S'yi  fi'^^i) ;
or, effecting the multiplications and replacing z^^Wx by x^y^y the values of x^y y,, 2^3, t^;.
contain the common foctor yj, which may be rejected. Also introducing on the left
hand sides the common £Eu;tor MM\ where Jlf = a/9 — 78, M^ = a')8' — ^'i\ the equations
become
MM'x^ = 7'&c, + OLd'yi — a'S^^i — a7'«!i,
MMy^ = /9i8'^+ 7S'yi  P^z^  i8'7«'i,
MMz^ = )87'iri + 7a'yi — /Sa'^Tj — 77'Wi ,
MMw^ = fflxx + aS'yi  SS'^j  dfi'w^,
^lues which give identically x^^ — z^^^ x^^ — z^w^. Moreover, by forming the value
of the determinant, it is easy to verify that the transformation is in fact an im
proper, ona We have thus obtained the equations for the improper transformation of
the surface xy^zw^^O into itself. By writing Xi + iyi, Xi^iyi for Xi, yi, &c., we have
the following system of equations, in which (a, 6, c, d), (a\ b', &, d/) represent, as
before, the coordinates of the poles of the plane sections, and Jlf' = a*H5* + c' + d*,
Jf'^«a'*H6'«Hc'*H(i^ viz. the system^
^ The system b very similar in form to, but is euentially different from, that which conld be obtained
^ the theory of qnatemionB by writing
the lasfcmentioned transformatiGn is, in &ot, j^roper, and not improper.
136 ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE [127
MWx^ = (oa'  hV  cc'  ddf) x^ + { ah' + a'b + cd'  c'd) yi
+ ( oc' + a'c + d6'  d'h) g:, + ( ad'+ a'd + he  6'c) w^,
iOfy, = (a6' + a'6 cd' + c'd) iCi + ( oa' + 66'  cc/  dd')y^
+ ( 6c' + 6'c  da' + d'a) ^x + ( 6d' + 6'd  oc' + a'c) w,
MM'z^ = (ac' + a'c  d6' + d'6) a^ + ( 6c' + 6'c ''ad'+ a'd) yi
{(aa' 66' + cc'  dd') ;?i + ( cd' { dd  6a' + 6'a) w,,
iOf'm, = (ad' + a'd 6c' + 6'c) (x\\{ 6d' + 6'd ca '\' da) y^
+ ( cd' {'dd a6'+ a'6) z^ + ( aa'  66' ''cd + dd') w^,
values which of course satisfy identically x^\y^\z^\w^ = x^\y^\z^ + w^y and which
belong to an improper transformation. We have thus obtained the improper trans
formation of the surface of the second order a^ + y' + 2:* + t(;* = into itself
Returning for a moment to the equations which belong to the surface xy'^zw = 0,
it is easy, to see that we may without loss of generality write a = ^=a' = )8'=0;
the equations take then the very simple form
MWx^^yBx,, MWy^^yB'yu M]irz^ = yy'w,, itf Jf' w, =  SS'^j,
where MJiT = V — 78 V — 7'S' ; and it thus becomes very easy to verify the geometrical
interpretation of the formulae.
It is necessary to remark, that, whenever the coordinates of the points Q are
connected with the coordinates of the points B by means of the equations which
belong to an improper transformation, the points P, Q have to each other the
geometrical relation above mentioned, viz. there exist two plane sections 0, <l> such
that P, Q are the opposite angles of a skew quadrangle upon the surface, and having
the other two opposite angles in the sections 0, <(> respectively. Hence combining
the theory with that of the proper transformation, we see that if A and B, B and
C, ..., M and N are points corresponding to each other properly or improperly, then will
JV and A be points corresponding to each other, viz. properly or improperly, according
as the number of the improper pairs in the series A and B, B and C, ..., M and N
is even or odd; i.e. if all the sides but one of a polygon satisfy the geometrical
conditions in virtue of which their extremities are pairs of corresponding points, the
remaining side will satisfy the geometrical condition in virtue of which its extremities
will be a pair of corresponding points, the pair being proper or improper according
to the rule just explained.
I conclude with the remark, that we may by means of two plane sections of a
surface of the second order obtain a proper transformation. For, if the generating
lines through P meet the sections 0, <(> in the points A, B respectively, and the
remaining generating lines through A, B respectively meet the sections ^, respec
tively in R, A', and the remaining generating lines through P', A' respectively meet
in a point P'; then will P, P' be a pair of corresponding points in a proper trans
127] OF THE SECOND ORDER INTO ITSELF, 137
formation. In fact, the generating lines through P meeting the sections 0, <f> m the
points Af B respectively, and the remaining generating lines through A, B respectively
meeting as before in the point Q, then P and Q will correspond to each other im
properly, and in like manner R and Q will correspond to each other improperly; i.e.
P and P* will correspond to each other properly. The relation between P, P' may
be expressed by saying that these points are opposite angles of the skew hexagon
PARP'A'B lying upon the surface, and having the opposite angles A, A' in the
section 0, and the opposite angles B, R in the section <(>. It is, however, clear from
what precedes, that the points P, P' lie in a section passing through the points of
intersection of 0, ^, and thus the proper transformation so obtained is not the general
proper transformation.
2 Stone Buildings, January 11, 1854.
c. n.
18
138
[128
128.
DEVELOPMENTS ON THE PORISM OF THE INANDCIRCUM
SCRIBED POLYGON.
[From the Philosophical Magazine, vol. viL (1854), pp. 339 — 345.]
I PROPOSE to develops some particular cases of the theorems given in my
paper, "Correction of two Theorems relating to the Porism of the inandcircumscribed
Polygon" {Phil. Mag, voL vi. (1853), [116]). The two theorems are as follows:
Theorem. The condition that there may be inscribed in the conic [7=0 an
infinity of ngons circumscribed about the conic F=0, depends upon the development
in ascending powers of f of the square root of the discriminant of ^U+V\ viz. if
this square root be
then for n = 3, 5, 7, &c. respectively, the conditions are
C=0,
C, D =0,
D, E
G, D, E
D. E. F
E, F,
= 0, &c ;
and for n = 4, 6, 8, &c. respectively, the conditions are
D=0,
D, E =0,
E, F I
D, E, F
E, F, G
F, G, H
= 0, &c.
128] DEVELOPMENTS ON THE PORISM, &0. 139
Theorem. In the case where the conies are replaced by the two circles
then the discriminant, the square root of which gives the series
il + £f + Cp + Df + ^f * + &c. ,
is
Write for a moment
^+£f+Cp + i)f> + ^f* + &c. = V(l + af)(l+6f)(l + cf),
then
^ = 1.
25 = a + 6 + c,
 8C = a» + 6^ + c« 26c  2ca  2a6,
162) = a» + 6» + c'a* (6 +c) 6^(0 + a) c»(a +6) + 2a6c,
128i; = 5a* + 56* + 5c*4a»(6 + c)46»(c+a)4c»(a + 6)
+ 4a»6c + 46»ca + 4c«a6  26»c»  2c»a»  2a>6«,
&c.
To adapt these to the case of the two circles, we have to write
r(l + af)(l + 6f)(l + cf) = (l+f){r»+f(r>+i?a«) + f'i?l,
and therefore
c = l,
values which after some reductions give
^=1.
r«.8C = (i?  a')"  4i2V,
r* . 162) = (i?  o») {(i?  o')'  2r» (i? + a%
r* . 128^ = 5 (i? o«)«  8 (iJ»o»)'(i? + 2r>)r»+ 16a«r«.
Hence also
»* . 1024 (GE  i)*) = {5 (i?  o«)*  8 (i?  o»)' (ii' + 2r») r» + 16aV} {(i?  o>)*  4i2V)}
 4 {(i?  o>)'  2 (ii»  o») (i? + a') r»j',
18—2
140 DEVELOPMENTS ON THE POMSM OF THE [128
which after all reductions is
+ 16J? (i? + 2a») (i?  a»)» r*
Hence the condition that there may be, inscribed in the circle a;' + y' — i? =
and circumscribed about the circle' (a? — a)" + y* — r* = 0, an infinity of ngons, is for
n = 3, 4, 5, Le. in the case of a triangle, a quadrangle and a pentagon respectively,
as follows.
I. For the triangle, the relation is
(i?  a»)»  4iJV* = 0,
which is the completely rationalized form (the simple power of a radius being of
course analytically a radical) of the wellknown equation
which expresses the relation between the radii R, r of the circumscribed and inscribed
circles, and the distance a between their centres.
II. For the quadrangle, the relation is
(i?a»)»2r»(J? + a») = 0,
which may also be written
(i2+r + a)(iJ + rtt;(i2r + a)(Bra)r* = 0.
(Steiner, Crelle, t. ii. [1827] p. 289.)
III. For the pentagon, the relation is
(R'''a''y  12i? (i? tt»)*r» + 16i?(i? + 2A^) (iPa»)"r*  64i?tt*r« = 0,
which may also be written
(iP  a^y {(i?  a^y  4iPr»}»  4iPr» {(iP  a^y  4a»r»}» = 0.
The equation may therefore be considered as the completely rationalized form of
(i?a*)> + 2iJ(i?a»)»r4i?(iPa»)r»8i2a»r«=a
This is, in fact, the form given by Fuss in his memoir "De polygonis symme
trice irregularibus circulo simul inscriptis et circumscriptis," Nova Acta Petrop. t. xiii.
[1802] pp. 166 — 189 (I quote from Jacobi's memoir, to be presently referred to). Fuss
puts iJ + a=jt>, R — a = qy and he finds the equation
jj^q^  r' (/)» + y*) _ / g  r
128] INANDCIRCUMSCRIBED POLYGON. 141
which, he remarks, is satisfied by r^^—p and r= ^ , and that consequently the
rationalized equation will divide by p + r and pq — r{p{q); and he finds, after the
division,
P^^+p^^ip + q)rpq(p + qyf^'(p\q)(P'qyr'=^0,
which, restoring for p, q their values R + a, R — a, is the very equation above found.
The form given by Steiner (CreUe, t. ii. p. 289) is
r (iZ  a) = (iZ + a) V(i2 T^M^) ~(^^r;^"^) + (22 + a^
which, putting p, q instead of iJ + a, B — a, is
qr =p 'Jip r){qr){p V(g  r) (g +p) ;
and Jacobi has shown in his memoir, "Anwendung der elliptischen Transcendenten
u. 8. w.," CreUey t. III. [1828] p. 376, that the rationalized equation divides (like that
of Fuss) by the factor pq'ip + q)r, and becomes by that means identical with the
rational equation given by Fuss.
In the case of two concentric circles a = 0, and putting for greater simplicity
. = Jlf, we have
il+£f + Cp + Df> + ^f* + &c. = (l + f)Vl + Jff.
This is, in fact, the very formula which corresponds to the general case of two
conies having double contact For suppose that the polygon is inscribed in the conic
(7=0, and circumscribed about the conic C7' + P* = 0, we have then to find the
discriminant of (U+U + P*, Le. of (l + f)Cr+P*. Let K be the discriminant of U,
and let F be what the polar reciprocal of U becomes when the variables are replaced
by the coefficients of P, or, what is the same thing, let — ^ be the determinant
obtained by bordering K (considered as a matrix) with the coefficients of P. The
discriminant of (l+f)i7+P" is (1 + f)»ir + (l + f)»P, Le. it is
(l+f)>{J5r(l+f) + P}, =(J5r + P)(l + f)>(l + iff),
K
^here M== j^ — jj,; or, what is the same thing, M is the discriminant of U divided
by the discriminant of U^P*, And M having this meaning, the condition of there
being inscribed in the conic (7 = an infinity of ngons circumscribed about the conic
^ + P* = 0, is found by means of the series
^+5f+6'f»+i)p + ^f* + &c. = (l + f)Vl+iff.
We have, therefore,
DEVELOPMENTS ON THE P0RI8M OF THE
A=l,
16/) = JP2Jtf',
~12HE = 5M*8M',
1024(C£D') = i£*(AP12if+16),
Hence for the triangle, quadrangle and pentagon, the conditions are —
I. For the triangle.
[128
IL For the quadrangle,
III. For the pentagon.
if + 2 = 0.
Jf 4 = 0.
and so on.
It is worth noticing, that, in the case of two conica having a ipoint contact,
we have F = 0, and consequently M=l. The discriminant is therefore (l+f)*, and
as this does not contain any variable parameter, the conies cannot be determined so
that there may be for a given value of « (nor, indeed, for any value whatever of
n) an infinity of ngons inscribed in the one conic, and circumscribed about the
other conic.
The geometrical properties of a triangle, Sk. inscribed in a conic and circum
scribed about another conic, these two conies having double contact with each other.
are at once obtained from those of the system in which the two conies are replaced
128] INANDCIRCUMSCBIBED POLYGON. 143
by concentric circlea Thus, in the case of a triangle, if ABC be the triangle, and
a, j8, 7 be the points of contact of the eidee with the inscribed conic, then the tangents
to the circumscribed conic at A, B, C meet the opposite sides BC, CA, AB in points
lying in the chord of contact, the linea Aa, Bff, Cy meet in the pole of contact,
and BO on.
In the case of a quadrangle, if ACEQ he the quadrangle, and b, d, f, h the
pointa of contact with the inscribed conic, theu the tangents to the circumscribed
conic at the pair of opposite angles A, E and the corresponding diagonal CQ, and
in like manner the tangents at the pair of opposite angles' C, Q and the corresponding
diagonal AB, meet in the chord of contact. Again, the pairs of opposite sides AC,
£6, and the line dk joining the points of contact of the other two sides with the
inscribed conic, and the pairs of opposite sides AO, GE, and the line hf joining the
points of contact of the other two sides with the inscribed conic, meet in the chord
of contact The diagonals AB, CQ, and the lines hf, dh through the points of
coDtact of pairs of opposite sides with the inscribed conic, meet in the pole of
^tact, &c
The beautiihl systems of 'focal relations' for regular polygons (in particular for
tlw pentagon and the hexagon), given in Sir W, B. Hamilton's Lectures on Quaternions,
[Dublin, 1853] Noa. 379 — 393, belong, it is clear, to polygons which are inscribed in and
tuwmscribed about two conies having double contact with each other. In foot, the focus
of s conic is a point such that the lines joining such point with the circular points at
"ifinity (Le, the points in which a circle is intersected by the line infinity) are tangents
^ the conic In the case of two concentric circles, these are to be considered as
''"Jelling in the circular points at infinity; and consequently, when the concentric
cucles are replaced by two conies having double contact, the circular points at infinity
*K replaced by the points of contact of the two conies.
144 DEVELOPMENTS OK THE PORISM, &C. [128
Thus, in the figure (which is simply Sir W, E. Hamilton's figure 81 put into
perspectiveX the system of relations
F. G{..)ABCI,
G. H{..)BCDK,
B,I(..)CDEF,
I, K(..)DEAG,
K, F(. .) EABH.
will mean, F, 0(.,)ABCI, that there is a conic inscribed in the quadrilateral ABCI
such that the tangents to this conic through the points F and pass two and two
through the points of contact of the circumscribed and the inscribed conies, and
similarly for the other relations of the system. As the figure is drawn, the tangents
in question are of course (aa the tangents through the foci in the case of the two
concentric circles) imaginary,
2 Stone Buildivgs, March 7, 1854.
129]
145
I
[
129.
ON THE PORISM OF THE INANDCIRCUMSCRIBED TRIANGLE,
AND ON AN IRRATIONAL TRANSFORMATION OF TWO TER
NARY QUADRATIC FORMS EACH INTO ITSELF.
[From the Philosophical Magazine, vol. ix. (1855), pp. 513 — 517.]
There is an irrational transformation of two ternary quadratic forms each into
itself based upon the solution of the following geometrical problem,
Given that the line
Ix \ my \nz =
Dtteets the conic
(a, 6, c, /, g, A$a?, y, zy =
in the point (o^, y,, Zi); to find the other point of intersection.
The solution is exceedingly simple. Take {a^, y^, z^) for the coordinates of the
other point of intersection, we must have identically with respect to a?, y, z,
(a, ...$a?, y, zy.(fSL, ...$i, m, wy&(te + my + mr)»
= (a,...$aH, yi, z^l^x, y, z).(a, ...Ja?,, y^, z^'^Xy y, z)
^ a constant foctor pris.
Assume successively a?, y, ^r = ®, ^, ffi ; ^, 98, Jp ; ffi, ip, Ct ; it follows that
C. n.
19
146 ON THE PORISM OF THE INANDCIRCUMSCRIBED TRIANGLE, [129
or, what is the same thing,
X.2 : y^ : Zt= yi^i (frw* + cfn^— 2/mn)
' ^1^1 (ctw' + &n* — 2hlm),
It is not necessary for the present purpose, but it may be as well to give the
corresponding solution of the problem :
Given that one of the tangents through the point (f , 17, f) to the conic
(a, 6, c, /, g, A$a?, y, zf =
is the line fia? f 7?i,y + ni^ = ; to find the equation to the other tangent.
Let Ijc + m^ + n^^ = be the other tangent, then
(a, ...$f, 17, t;y.{a,...\x, y, zf [{a ...J^, 17, t$a?, y, z)Y
to a constant factor prh. Assume successively y = 0, ^ = 0; £^ = 0, a? = 0; a: = 0, y = 0;
then we have
i, : m, : 71,= miWi {a (a, . . . $f , 17, ?)"(af + Ai7+5rJ)»}
: n,Z,{6(a,...$f, 17, ?)»  (Af + 617 +y?yi
: iim,{c(a,...$f, 17, r)»((7f+/^ + cO'};
or, as they may be more simply written,
i, : 771, : n,= 7/^1^ (J8(r* + ®i7« + 2JP17O
: 71, fi (ODp + ia?»  2® tf )
Returning now to the solution of the first problem, I shall for the sake of
simplicity consider the formulae obtained by taking for the equation of the conic,
«^ + /Sy* + 78^ = 0.
We see, therefore, that if this conic be intersected by the line lx\my + nz=^Q in
the points {x^, y,, z^ and {x^, y„ j?,), then
«j : yj : ^a = yiZi (77?i» + a/«»)
: ZiX^{an^ \fifi)
129] AND ON AN IRRATIONAL TRANSFORMATION &C. 147
We have, in fact, identically
lyi^i (^w* + 7W1") + mZiXi (yP + cm*) + rw^i (am* + 0P)
= {amnxi + firdyi + ylmzi) {Ix^ + myi + nz^) — J?/m {ax^ + /9yi' + 72^1').
«yi V (/9»" + 7*^")" + fiziW (yl* + an*)« + 7^1 V («^' + ^^')"
= a/97 {  PiTi' — m'yi* — n'^i*
+ (twyi + nzi) Pa?i* + (n2:, + Ix^) mh/^ + (Za^ + my^ n^y^  Zlmnx^y^Zi] {Ix^ 4 wy, + /l^i)
 (I'fiyxi" + w Vyi' + n^fiz,') (aci* + )9y,« + 7V) ;
which show that if Ixi^myi^nZiO and aa?i» + )9yi* + 71^* = 0, then also lxi^mya + nzi =
and OiT,* + /Sya' + 7er,* = : this is, of course, as it should be.
I shall now consider Z, m, n as ffiven functions of Xi, yi, Zi satisfjdng identically the
equations
Ixi +myi +nzi =0,
IHk + m^ca + n^b = 0,
equations which express that lx + my + nz = is the tangent from the point (xi, y^, Zi)
to the conic cur" + 6y* + cz^ = 0. And I shall take for a, /9, 7 the following values, viz.
a = ax* + by I* + cz^*  a (a?i* 4 y^* + ^1*) ,
/9 = cuTi* + byi* + cgTi*  6 (x^* + y,* + ^1*),
7 = cur,' + 6yi' + c^i'  c (iTi* + yi* + 2^1*);
80 that iTi, yi, ^1 continuing absolutely indeterminate, we have identically ax^k ^yi^ + yz^ = ^.
Also taking as a function of Xi, y^ ^,, the value of which will be subsequently
pven, I write
X2 = ©yi^i (fin* + 7m»),
y2 = ^ZjXi(yP +an%
Z2 = Sx^i(am*+l3t');
^ that Xi, yi, ^1 are arbitrary, and x^, y^, z^ are taken to be determinate functions
^^ *i> yi, 2?!. The point (arj, yj, ^j) is geometrically connected with the point (xi, y^ z^)
^ follows, viz. (a?a, yj, ^,) is the point in which the tangent through (xi, yi, z{) to
the conic cur" + 6y* + c^* = meets the conic passing through the point (a^, yi, ^Tj) and
tie points of intersection of the conies ax* + b^ + cz* = and a:* + y* + ^' = 0. Con
^uently, in the particular case in which (a?i, yi, ^1) is a point on the conic
^+y'+^ = 0, the point (x^, ya, ^2) is the point in which this conic is met by the
t^gent through (a^, y,, Zi) to the conic ax^ + by* + cz* = 0,
It has already been seen that Ix^ + myi + n^i = and aa?i^ + /9yi* + 7^1' = identically ;
consequently we have identically Ix^ 4 my^ + w^Tj = and ouv^* + fiy^^ + 7^2' = 0. The latter
equation, written under the form
(ax^* + by,* + cz*) {xf + y,' + z^")  (iCi» + yi» + ^1') (cw?, + by.^ + c^aO = «,
19—2
148 ON THE PORISM OF THE INANDCIRCUM80RIBED TRIANGLE, [129
shows that if a^^ y„ z^ are such that a?j' + yj' + ?,• = a^' + y^' + ^j', then that also
aoi^^ + by^ + cz^=^axi* + byi* + czi\ I proceed to determine O so that we may have
^i* + yi' + V = ^' + yi' + V« We obtain immediately
^,(^«' + y«' + '^>') = (^V + mV + »V)(aV + i8»yi« + 7«^i«)
write for a moment
CM?i* + 6yi' + C2r,'=p, iTi' + yi' + 2^1* = J, so that a=*p^aq, fi^p — bq, y^^p — cq,
then
o»ah' + I3h/,^ + T*^,' = 2p*  2p .;)5 + (aV + 6»yi« + (fz^*) (f,^q {(aV + ^V + c»^i') 3 i>*}.
= J {(6  c)« yiV + (c  a)» 2r,V + (a  6)' fla'y,'},
a?iV + i8*wiV + Y*»iV  2fiymWyi%*  2yan^l^zW  2a/8i»m'j?i«yi«
— 2pg {oZ^ar/ + fcm^y/ + c?i*V — (6 + c) rnWy^z^  {c + a) nH^z^x^ — (a + 6) 1hii^x^y^\
the first line of which vanishes in virtue of the equation Ix^ + myx + w^i = ; we have
therefore
^(a^'+y,' + ?,•) H (^1* + yi» + z^^)
= (Z V + ^V + wV) {(6  c)» yi V + (c  a)» Vj?i* + (a  6)» iCj V}
+ 2 (aa?i* + 6yi' + cz^^) {aZV + &^V + cn%^(b + c) mWy^z^ (c + a) nH%W  (a + 6) ?mV,*yi»}
 (^' + yi' + 8^1*) {a'^V + 6*m^/ + c^V  26cm«n«yi V  ican^l^z^x^  iabl^Wyi^]
Hence reducing the function on the righthand side, and putting
we have
+ (c*wi*  26«m V) yi V + (cM  2c»n*P) ^i V + ( W*  2a«Pm«) aj^ V
+ (bW  2c*mV) yiV + (c»Z*  2aWZ«) ^^ V + (aHn*  26»Z>m«) iCj^y/
+ ZmW {bcca''ab) + 2n'Z' (6c + caa6) + 2Pm* (  6c  ca + a6)} Xi%%\
The value of might probably be expressed in a more simple form by means
of the equations Ixi + myi + n^i = and Z'6c + m^ca + n*a6 = 0, even without solving
these equations; but this I shall not at present inquire into.
129] AND ON AN IRRATIONAL TRANSFORMATION &C. 149
Recapitulating^ I, m, n are considered as functions of ^, y^, ^i determined (to a
common £EU^tor pria) by the equations
i*a + wiyi +nzi =0,
Pbc + rn^ca + n^ab = ;
B is determined as above, and then writing
we have
y, = ©^iri(7p +an'),
^, =5 ©a?iyi (am* + fiV) ;
and these values give
Is^ +my, + n2r, = 0,
xf +y,' +?,» =a?i« +yi' +zx\
axf + tyt* + ozf = cuTi' + 6yi' + c^i*.
In connexion with the subject I may add the following transformation, viz. if
Z^Jaaf^ V3/8 (y  f) + V(3a  2/8) (dj» + y» + z") + 2/8 (y;? + ;?a? + «y),
f
then reciprocally
3>//8a?=rV3a(y'/) + V(3/82a)(a/» + y'» + /«) + 2a(y'/ + /«^ + fl?'yO.
a^ + y» + ^ =a?'» + y'« + /«,
/8 (aj* + y* + jt* yz^  ^a?  icy) = a (a^» + y'* + /' y'^'  2'a?'  a?'yO.
Suppose 1 + /:> + p^ = 0, then
a^ + j^{2^ — yz'zx — xy = (x + py + p^z) (a? + p* + p^) ;
and in feet
3Va(a?' + py'+p»/) = >/3)8(l + 2p)(a? + py + p«^),
3Va(a?'+py+p^) = V3i8(l + 2p)(a? + p»y + p^).
The preceding investigations have been in my possession for about eighteen months.
2. Stone Buildings, April 18, 1855.
150
[130
130.
DEUXIEME MEMOIRE SUR LES FONCTIONS DOUBLEMENT
PERIODIQUES.
[From the Journal de Maihimatiques Pures et Appliques (Liouville), torn. xix. (1854),
pp. 193—208: Sequel to Memoir t. x. (1845), 35.]
Je vais essayer de d^velopper ici les propri^t^s qui se rapportent aux transformations
lin^aires des p^riodes des fonctions yw, gx, Gx, Zx, dont je me suis occupy dans le
M^moire sur les fonctions doublement p^riodiques que j*ai donn^ dans ce Recueil en
1845. Avant d*entrer en matifere, je remai^que que partant des expressions
des deux p^riodes, oi i = V — 1, on obtient, en ^crivant
fl* = ck) — a)'i,
T»=i;i/i,
les Equations
ft*T = wu + G) V + i (W  w'u),
au moyen desquelles et des valeurs
IIT mod. {(OV  do'v) ' SIT mod. {mv  (o'u)
des quantity fi, B, on ddduitles formules
ft mod. {(OV — a)'i;) ' T mod. («t;' — (o'v) '
130] DEUXltlfB ICl^OIRE SUB LES FONCTIONS DOUBLEMENT PilBIODIQUES. 151
Je ne &is attention qu'aux transformations qui correspondent k des entiers impairs et
premiers, et je suppose, de plus, que la transformation soit toujours propre et r^gulifere ;
c'estidire qu'en ^rivant
(2* + 1) ft, = xn + mT = (X, /Lt),
(2A + l)T, = i;ft + pT = (i/, p),
oh 2k +1 est un entier positif, impair et premier, et oh X, fi, v, p sont des entiers
tels, qu'au signe pr^ \p^fiv soit 6gsA k 2A; + 1, je suppose
Xp — fiv=2k + 1,
(condition pour que la transformation soit propre), et, en outre,
X= 1, /iA = 0, (mod. 2)
1^ = 0, p=l,
(condition pour que la transformation soit r^guli^re).
On trouve tout de suite
ft= pn,/tT, = (p, /^X,
T=./n, + XT, = (i., XX;
j'^cris aussi
ft, « tt), + ©/», ft,* = tt),  «/»,
et je suppose que i3„ /3, soient des fonctions de o>,, v, telles que les fonctions B, 0,
de 0, V.
Cela ^tant, je forme d'abord T^uation
(2* + 1) (o),v/  o)>,) = 0)1/'  w't;,
au moyen de laquelle T^uation
^ ^^ mod. (a)i/  G)'i;)
8e transforme en
Beli
i<"^')(^^^>^i" „,.d(.;.>.) {°°^=nf^}'
mod.
mod.
9rt(<»,v/o)>,)
~ nX mod («,v/  «>,) ^"' "^ '* * "' •
152 DEUXlklCB M^OIBE SUB LES FONCTIONS DOUBLEMENT P^RXODIQUES. [130
oa enfin
et de mSme
{2k + l(Bfi)B,]a^^fi^(p, X);
Equations qui seront bientdt utiles.
Je suppose d'abord que 2k + 1 soit ^gal k Tunit^, transformation que Ton peut
nommer trivicUe, La fonction yx est d^finie par I'^quation
ya? = e**^a?nl + ^^^J, mod.(m, n)< T, r= oo;
dans (m, n) = mCl + nT, les entiers m, n doivent prendre toutes les valeurs positives
ou negatives (le seul syst^me m = 0, n = except^) qui satisfont k Tin^galit^
mod. (m, n) < T,
dont le second membre T sera ensuite suppose infini. Soit y,x la fonction corres
pondante pour les p^riodes fl^ , T^ ; on aura
V ar = €**.«• aril l +7^.1, mod.(m, ny<r, r=«.
Or
= m (Xil + fiT) + n (i/fl + pT),
= (Xm + vn)il + (jjLfn + pn) T,
= m^fl + n,T,
En ^rivant, comme nous venons de le faire,
m, = Xm + im,
on voit tout do suite qu'ii chaque systfeme de valeurs entiferes de m, n, correspond
un Hystfemo, ct un seul systfeme, de valeurs enti^res de m^, w,; et que de m6me k
chaque syst^me de valeurs entiferes de m,, n,, correspond un systfeme, et un seul syst^me,
do valeurs untiferes do m, w ; de plus, les systfemes m = 0, n = et m, = 0, n^ = 0,
correspondent Tun k Tautre. II est done permis d'^rire
( (m, n)j ( (m, n),)
les limitoH comme auparavant ; car, k cause de
(m, n), = (m„ n,%
gx
130] DEUXrfeMB Ml^OIRE SUR LES FONCTIONS DOUBLEMENT PiaaiODIQUBS. 153
la condition pour les limites, savoir :
mod (m, n\ < T, T= oo ,
devient
mod. (m, n) < r, T = oo .
Cela donne enfin I'^uation
et, au moyen de cette Equation, on obtient une ^nation correspondante pour la trans
formation de Tune quelconque des fonctions yx^ gx, Ox, Zx, d^finies par les ^nations
yar = 6**^.a;nl+^^^^, mod. (m,n)<r,
= e**^. nil 4^^^}, mod. (m, n) < T, r=oo;
Ox^e^^. n(l+7^J, mod. (m, fi) < r,
\ (m, n)j
Za? = €**^. u\l+'JX, mod. (m, n) < r,
(^nations dans lesquelles m = m + ^, n = n + ^). Je prends par exemple la fonction gx,
et j'&ris dans I'^uation entre y^x et yx, x + ^fl au lieu de x. Soit pour un moment
p = 2p' 4 1, /A = 2fi\ cela donne
a?+jn = ar + i(/)ft,A*T,)=a? + (p',/).
Done
y, (x + ill) = e^.*'P. '*')' M^g^x,
c'estidire
y, (a? + ifl) = 6*^'*<^» **>' if,g,aj ;
de plus,
y (x + ill) = 6*^0* ifgar.
Ces substitutions ^tant eflFectu(?es, les coefficients M, M, doivent 6tre ^limin^s en ^rivant
««0; cela donne
ou enfin, au moyen d'une ^nation ddji trouv^,
et de m^me pour les fonctions Ox, Zx,
c. n. 20
154 DEUXlilME ICl^OIRE SUB LES FONCTIONS DOUBLEMENT P^RIODIQUES. [130
Done eofin, en reprdsentant par Jx Tune quelconque des fonctions jx, gx, Ox, Zx,
on aura
oil J,x est ce que devient Jx au moyen d'une transformation triviale (propre ct
r^gulifere) des p^riodes.
Je passe k pr^nt k la transformation pour un nombre impair et premier {2k + 1)
quelconque; mais pour cela on a besoin de connattre la valeur de la fonction
"' = "{^^ (m, r)+y }' ^od.[(m.n) + y}<T, r=oc.
o{i y = a + bi est une quantity r^elle ou imaginaire quelconque.
Soit u ce que devient u' en prenant pour la condition par rapport aux limites
mod. (rw, n) < T, 7= oo ;
on trouve sans peine
y(y)
Pour trouver u\ je forme T^uation
u : u' = u\l 4 . ^ J ,
la limite inf^rieure du produit infini double ^tant
mod. {(m, n) + y]> T,
et la limite sup^rieure
mod. (m, n) < T, T= x ;
cela donne
logttlog«'=..S(^;i7^i^S,(^, „\+y}.+ .
car on peut d^montrer que
S 7^=0. ^7^=0, &c.
■^^ (ot, n)* (m, n)*
Pour cela, observons que m et n ^tant infinis puisque T Test, la premifere des sommes
dont il s'agit peut se remplacer par rint^grale double
J __ rCdmdn
130] DSUXI^ME M]^OIB£ SUB LES FONCTIONS DOUBLEMENT PilBIODIQUES. 155
laquelle (en Ajrivant m = r cos d, n = r sin ^, ce qui donne, comme on sait, dmdn = rdrdff)
devient
//
drd0
r{ilcos0{'r8ui0y'
rfoti
(log r) dd
II
(aco&e+rQindy
en prenant (logr) entre les limites convenablea Pour trouver ces limites, j'^ris
(m, n) + y = r(ncos^ + TsiD^) + y;
ce qui donne
mod.' {(m, n) + y} = {r (flcos ^ + T sin ^) + y} {r(fl* cos d + T*8in 0) + y*},
fiavoir, k Tune des limites
r»(ftcos^ + T8ind)(n*co8^ + T*8in^)
+ r{y*(nco8^ + T8in^) + y(ft*cos^ + T*sin^)} + 2* = 0;
ou, en n^gligeant les puissances negatives de T,
T
r =
V(n cos h T sin 0) (ft* cos ^ + T* sin 0)
1 f y y* I
"*t"cos^ + T8in^"*"ft*cos^ + T*8in^j'
et it Tautre limite,
r =
V(ft cos e + T sin 0) (ft* cos ^ + T*sin ^) *
Or, en repi^sentant ces deux ^nations par
?• = 22 — <^, r = 22,
on trouve, pour la valeur de (log r) entre les deux limites,
logi2log(i2^) = log(l) = 0,
i cause de la valeur infinie de 12. Ainsi la somme cherch6e est nuUe ; et il est tout
clair que les sommes suivantes S / ^ » &c., se r^uisent de m6me k z^ro.
(m, n/
Done enfin,
20—2
156 DEUXifeME MilMOIRE SUR LES FONCTIONS DOUBLEMENT PilRIODIQUES. [130
Cela fait voir que
u' = e"** M,
le coefficient k ^tant donn^ au moyen de T^uation
o{i la somme est prise, comme auparavant, entre les limites
mod {(m, n) + y} > T, mod. (m, n) < T, r=oo.
Mais il n'est pas permis d'^rire
J J (m, n)
En eflTet, cette intdgrale n'est que le premier terme d'une suite dont il faudrait, pour
obtenir un r^ultat exact, prendre deux termes; le second terme de la suite serait
une int^grale prise le long d'un contour, et il serait, ce me semble, tresdifficile d en
trouver la valeur. Pour trouver la valeur de A, je remarque que k sera fonction
lin^aire des quantity T^ y, y*, 4f , &c., qui entrent dans les valeurs de r ; done,
puisqu'en demifere analyse I'=oo, k ne pent 6tre que de la forme Ly { My*, Cela
^tant, en substituant pour u' sa valeur, je forme T^uation
y(^+y) _ g_j«,t g(.si/+Ly+iry)« n II +  1
y(y) * I {rn,n) + yy
mod. {(m, n) + y]<T, r= oo ,
et j'&ris successivement
ce qui donne pour les valeurs correspondantes du produit infini double e"**** . gx et
€~****. Ox ; en comparant les valeurs ainsi obtenues avec les ^nations qui donnent les
valeurs de y(a? + in), y(af + iT), on trouve
2i = 0, M^ '^
mod. («i/ — oi't;) *
ou enfin,
y(a: + y) ^ ^_^^ /^^mod. (l>v)^)^ n(l+— ^— I
y(y) ' 1 Kw) + yJ*
mod. {(m, n) + y} < T, r= oo ,
laquelle est I'^uation qu'il s'agissait d'^tablir. II est & peine n^essaire de faire la
X
remarque que pour y = 0, on doit consid^rer & part le fiawteur 1 +  , lequel multipli^
if
par y(y) devient tout simplement x\ T^uation subsiste done dans ce cas.
130] DEUXIjkaiE Ml^OIBE 8UR LES FONCTIONS DOUBLEMENT PilRIODIQUES. 157
En revenant au probl^me des transformations lindaires, partant des ^nations
(2*; + l)ft, = Xfl+/AT,
(2A: + 1)T, = i/n+/)T,
je suppose d'abord que les coefficients X, v ne satisfassent pas k la fois aux deux
conditions
X = 0, 1/ = 0, mod. (2*; + 1),
etje prends p, q des entiers quelconques tels, que \p\pq ne soit pas =0, mod. (2A; + 1).
Cela ^tant, soient
\p + vq =;>„
fMp + pq =gr„
(2*: + l)t=i>/"+3.T,
et, par cons^uent,
Je forme T^uation
savour
c'estridire
\m + im — «p, = (2k f 1) m^,
fjLm + i/n — sq, = (2A; + 1) n^ ,
ou, ce qui est la m^me chose,
n ^ sq = m^ fjL  n^X.
Or, m^, n^, ^ ^tant des entiers donn^, m, n seront aussi des entiers; de m6me, m, n
^tant des entiers donnas, on trouve de A: i — i un entier a qui donne m^ un entier.
V&is cela ^tant, n, sera aussi un entier; car autrement n^ serait une fraction ayant
pour d^nominateur, lequel on voudrait, des nombres 2& + 1, X, i/, ce qui est impossible
i moins que
X = 0, V = 0, mod (2A; + 1).
^^ si ces ^nations avaient lieu, on trouverait d'abord 8 de mani^re h, avoir n, entier,
^t alors, puisqu'on n'a pas aussi
/A = 0, p = 0, mod. (2* + 1)
(^u effet, cela est impossible k cause de T^uation Xp — /ty = 2i + 1), on d^montrerait,
«>nmie auparavant, pour n^, que m, est entier. Done, enfin, rw, n ^tant des entiers
doim&, on trouve pour m,, n,, « un systfeme d'entiers tel que 8 soit compris de A: it — A,
«t Von voit sans peine qu'il n'y a qu un seul systfeme de cette espfece.
158 DEUXltME MISmOIBE SUB LES F0NCTI0N8 DOUBLEMENT pArIODIQUBS. [130
A pr&ent, partant de T^uation
y (y) I K. w,) + yj
(et faisant attention k la particularity que prints le cas de y = 0), j'^ris successivement
y0. y = ±<^,..., y = ±h^,
et je forme le produit des ^nations ainsi trouv^e& Cela donne, k cause de (tn,, n,) + «^
= (m, n),,
y(»f) I (»». «)/J
la condition, par rapport aux limites, ^tant
mod.(m, n),<T, T=ao.
Or
avec la m^me condition, par rapport smx limites; done, enfin,
y^,.,*t.,^.«^..ny^>,
oil, dans le num^rateur, s doit avoir toutes les valours enti^res depuis s^ — k jusqu'i
« = + A:, y compris « = 0, et dans le d^nominateur ces m6mes valeurs, hormis la valeur
8 = 0.
II est, k present, facile de faire voir que cette propri^t^ subsiste pour I'une
quelconque des fonctions yx, gx, Ox, Zx\ en eflTet, pour la d^montrer pour gx, jecris
X + ^11 au lieu de a? ; en prenant, pour un moment, p = 2p' + 1, /a = 2/a', cela donne
c*estidire
y, (*fi)
Or, on d^uit de I'expression pour y^x,
■ y,(ifi)
y>+Jii) = eW.«(P.M),g^a,.
= etf ,» (P. M) , w (B,5+u) (*»+ite) n y (^ + *^ + i")
= eV.<* (p. M), £4 (»,5+i«)ir« ji dis+i) (Mte ij g (^ + ^Y*) .
130] DBUXltME ICl^OIRE SUB LES FONCTIONS DOUBLEMENT P^RIODIQUES. 159
ou enfin, k cause de T^uation
la valeur de g^x est
a a; = €* (*.^+^^«« . n g (^ + y .
»' g («V^)
et en repi^sentant, comme auparavant, Tune queleonque des fonctions yx, gx, Gx, Zx
par Jx, on a T^uation
equation dans laquelle 8 doit avoir, dans le num^rateur, toutes les valeurs entiferes
depuis « = — i jusqu'i 8 = k, y compris « = 0, et dans le d^nominateur, ees mSmes valeurs,
hormis la valeur « = 0.
Je suppose que les valeurs de p„ q, soient donn^es (cela va sans dire que Ton
ne doit pas avoir k la fois p^sO, j, = 0, mod.2A: + l), et je remarque que Ton a, pour
determiner X, fi, v, p, les conditions
pp,  vq^ = 0, mod. {2k + 1),
X=l, /A=0, mod. 2,
v=0, p=l,
Xp — fiv = 2k\ 1.
^^ cela ^tant, on aura ensuite, en rassemblant toutes les Equations qui ont rapport k
la transformation,
Pi>/ »'?/ = (2* +!);>,
(2A + l)n, = Xft +mT,
(2A + l)T,= i/ft +pT.
160 DEUXltME liiafOIRE SUR LES FONCTIONS DOUBLEMENT P^UUODIQUES. [130
Or, quoique les valeurs de X, /i, v, p ne soient pas compl^tement d^termin^es au
moyen de ces conditions, cependant il est clair que la valeur de la fonction J^x ne
depend que des valeurs de p„ q, (en eflfet, ces valeurs suffisent pour determiner la
quantity '4^=p^fl + gr^T, de laquelle depend la fonction J^x). Les formes diflKrentes de
J,x, pour les systfemes de valeurs de X, ft, v, />, qui correspondent k des valeurs
donn^es de p,, q,, doivent done se d^river de Tune quelconque de ces formes, au moyen
dune transformation triviale des modules 11^, T^. II est, de plus, clair que les valeurs
de p,, q,, qui sont ^gales k des multiples de (2A; + 1) pr^, ne donnent qu'une seule
valeur de J,x. Je suppose d'abord que
p, = 0, mod (2* + 1),
on pent trouver un entier tel que
0p,^l, mod.(2A: + l);
en prenant alors
0q, = q,, mod(2A? + l),
cela donne
(p,a + q,T,) = ft + qX mod. (2* +1),
savoir
^ = ft + qX Mttod. (2*; + 1).
Mais en donnant k 8 des valeurs entiferes quelconques, depuis — k jusqu'& k, le syst^me
des valeurs de syjt est equivalent au syst^me des valeurs de 80^, mod.(2A;+l); il est
done permis d'^crire, sans perte de g^n^ralite,
^ = ft + g,T.
De m^me pour
p, = 0, mod. (2k + 1),
on d^montre que Ton pent donner k q, une valeur quelconque, sans changer pour
cela la valeur de J^x\ il convient d'avoir p^ impair et q^ pair. J'^cris done, pour le
premier cas, 2q^ au lieu de g^, et je suppose que, dans le deuxi^me cas, les valeurs
de p,, q^ soient
p, = 2A: + l, 5, = 2.
Cela donne:
Premier ca8.
^ = ft + 2q,T,
q^ un entier quelconque, y compris z^ro, depuis —A? jusqu'^ +A:.
DeuxUme ca8.
^ = (2ifc + l)ft + 2T;
le nombre des valeurs diff^rentes de "9 sera done, en tout, 2A; + 2.
130] DKUXliOIB M^OIBE SUB LES FONCTIONS DOUBLEMENT P^BIODIQITES. 161
On obtient tout de suite, pour le premier cas, le syst^me d'^uations
X = l, /*= 2q„
p = l, 5 = 0;
^ = 2A+l(" + 2q,T).
Le cas particulier le plus simple est eelui de 9^ = 0; cela donne
^ = "' = 2lTl"' '^' = '^'
et, de 1^,
11
1 n
T, 2ifc + l T'
et m^me le cas g^ndral se r^uit k celuici, car, au moyen d'une transformation
irimale, on obtiendrait
ft' = n + 2q,T, T' = T.
et puis
Vr = ft,=
2k + l
ft', T, = r,
et, de ]&,
1 ft'
ft
T, 2k + lT'
Les ^nations correspondantes pour le deuxi^me cas sont:
p, = 2k+l, 9, = 2,
\=2A+1, /t=0,
V =0, p = 1,
p=l. 9=2,
^ = 2*1:1 t^^* "^ ^^ " "^ ^^
ft, = ft.
T =
' 2k + l
T;
^ qui donne
c. n.
21
162 DEUXltME MJ^OIBE SUB LES FONCTIONS DOUBLEMENT PI^ODIQUES. [130
J'ajoute, sans m'arrSter pour les d^montrer, quelques formules de transformation
pour le nombre 2; je trouve d'abord
Ces ^nations donnent, en introduisant les fonctions elliptiques, <f)x, fx, Fx donn^ au
moyen de
, yx  gx „ Gx
^=fc' >=k' ^^=:^'
les ^uations
F,x ^ Fx '
dont la seconde peut encore s'^rire sous la forme
et les deux ^nations combin^es ensemble conduisent sans peine & la valeur des
modules c^, 6^. On trouve en effet, en mettant comme & Tordinaire 6' = c' + 6*,
c/ = 46c,
6/ = (6c)»,
et puis
 1 — c (c — 6) i?x
*'^= Wa; '
 _ 1  c (c h 6) ^'a ?
•^'^■■lc(c6)<^«a?'
F.x=^
lc(c6)^fl?'
formules qui correspondent & celles de la transformation de Lagrange. Les ^nations
pour y^a?, Z^x donnent encore une valeur de ^^a?, laquelle, ^gal^ k la valeur qui vient
d'etre trouvfe, donne
yx%xZ'^(\iX) _ if>xfx
Z{x''\il)Z{x + Jft) " 1  c(c6) <^«a? *
130] DEUXliaCE M^OIRE SUB L£S FONCTIONS DOUBLEMENT P^BIODIQUES. 163
On obtient tout de suite les formules pour la transformation analogue il, = il, T^ = ^T.
Mais il &ut de plus consid^rer le syst^me
on aura alors
et puis, en Anrivant
on obtient
ll, = i(nT), T, = i(n + T):
r yx = €~* <*'*^^ gX Zx,
^_,...«.^y (^+inT)y(54nT)
^' yant)
z* ( jn  T)
y»(in + T)
^» (in + T)
c,» = (c  tc)«,  e; = (e+ icy,
Ax
^'* ^fiTF'x '
, 1 +ice4^x
J'" ~ fxFx '
ltce4fx
l + tce<ya; Z(a; + inT)Z(a;iflT)
fx Fx Z* (iir^) gx Ox
lice^fx Z(a! + ifl + T)Z(a;^ + T)
fxFx Z*{^ + t)gxOx
\\iee^x ^ Z{x + \€lT)Z{x^T)Z>(^ + r) ^
\ice4fx Z {x + ^oTT) Z (x  ^T^t) Z* {^ T)'
01, an moins, ces formules seront exactes au signe de i prfes; car il serait peut6tre
*"ffi<ale de determiner quel est le signe qu'on doit domier h, cette quantity.
21—2
164
[131
131.
NOUVELLES RECHERCHES SUR LES COVARIANTS.
[From the Journal filr die reine und angewandte McUhematik (Crelle), torn. XLvn. (1854),
pp. 109—125.]
Je me sers de la notation
(oo, ai,...an)(a?, yT
' pour repr^enter la fonction
en supposant que lea coefficients ao\ a^ &c. soient donnds par T^quation
(oo, Oi, . . . On) (X^ + AAy, \'x + fiyY = (oo', Oi', . . . O (a?, y)\
suppose identique par rapport k x, y, soit ^(oo, ai,...an\ x, y) une fonction des co
efficients et des variables, telle que
^K', Oi', ...On'; X, y) = (X/X»P^(ao, ai,...an; \x + fiy, \'x'\fify)]
cette fonction ^ sera gdn^ralement un Covariantt et dans le cas particulier oil if> est
fonction dea seuls coefficients, un Invariant de la fonction donn^.
Je suppose d'abord que les nouveaux coefficients soient donn^ par T^uation
. (Oo, Oi, ... an)(x + \y, y)~ = (ao', a,', ... an')(x, y)«;
cela donne les relations
Oi' = Gti + XOo,
Oa' = ttj + 2\ai + X'Oo,
&c.
131] NOUVELLES BEGHERCHES SUB LES COVABIANTS. 165
n £Biut done que le covaria/nt ^ satisfasse & T^uation
4>W> <^> ... On'; a?, y) = <^(ao, Oi, ... a„ ; a? + Xy, y),
laquelle peut aussi Stre ^rite comme suit:
4>W> ai',...a„'; a?Xy, y) = 0(ao, Oi, ...On; a:, y). (Z)
De mSme, en fiEusant
(Oo, Oi, ...a„)(a?, fix + yy^ioo", (h\...an")(x, y)«,
ce qui donne
On' =an
le cooaruint ^ doit satisfaire aussi & I'^uation
<t>(ao\ <h\'"(hi'\ a?, /iic + y) = <^(ao, Oi, ...a„; «, y); (F)
et r^proquement, toute fonction ^ homog^ne par rapport aux coeffieients et aussi par
rapport aux variables, qui satisfait k ces ^nations (X, Y), sera un covariant de la
fonction donn^.
Examinons d'abord T^uation (X) que je repr&ente par ^' = 0. Soit pour le
moment, Oi'— ai = Xai, a,' — a, = \aa, &c., alors on aura, comme k Tordinaire, T^quation
symbolique
oh les quantity Oi, Os, &c., en tant qu'elles entrent dans cti, a,, &c., ne doivent pas
6tre affect^ par les symboles da^, 3a,, &c. de la differentiation. En substituant les
valours de a^, a,, ...,et en ordonnant selon les puissances de \, cette ^nation donne
ah les symboles D, Di, &c. sont donn^ par
Q = ao3«, + 2aiaa, . . . + naj^ida^,
et les quantity Oi, a,, &c., en tant qu'elles entrent dans les symboles Q, Di> &c, ne
doivent pas Stre affect^ par les symboles da,, do,, &c. de la differentiation. II est
assez remarquable que T^quation symbolique peut aussi 6tre ^rite sous la forme plus
simple
0' = e^Dy^.) <^,
166 NOUVELLES RECHERCHES 8UR LE8 CO VARIANTS. [131
oh lea quantit^s Oi, a,, ..., en tant qu'elles entrent dans le symbole D, sont cens^es
affect^s des symboles do,, d^,, &c. de la diff(^rentiation ; de mani^re que dans le d^veloppe
ment, D'.0 par exemple, signifie D . D0, et ainsi de suite. Je ne m'arrSte pas sur
ce point, parce que pour ce que je vais d^montrer de plus important, il suffit de faire
attention a la premiire puissance de X. D'ailleurs Tintelligibilit^ des ^nations dont
il s'agit, sera facilitde en faisant les d^veloppements et en comparant les puissances
correspondantes de \. Cela donne par exemple:
n«=g«+2n„ n»=n»+3nni+6n„ &c.
oil les symboles D^ D^ &c. k gauche de ces Equations d^notent la double, triple, &c.
rdpdtition de reparation D, tandis qxx'k c6t^ droit des Equations, les quantity Oi, a,,... &c.,
en tant qu'elles entrent dans les symboles D, Di, &c. sOnt cens^ ne pas Stre
affect^es des symboles da^, do,, &c. de la diff(^rentiation. Dans la suite, si le contraire
n'est pas dit, je me servirai des expressions D', D", &c. pour d^noter les repetitions de
rop^ration, et de m6me pour les combinaisons de deux ou de plusieurs symboles.
Cela etant, T^quation 0' = c^(Di'»J ^ = donne
<^={i+x(nya,) + j^*2^nya,)»+...}0,
oil (□ — y3«)'.0 (JQ le r^pfete) equivaut a (D ""y3«)'(g ~y^«)^> ®^ *"^ d® suite. H
faut d'abord que le coeflScient de X s'^vanouisse, ce qui donne (D— y3jB)^ = 0; et cette
condition ^tant satisfaite, les coefficients des puissances sup^rieures s'^vanomssent d'elles
mSmes ; c'est&dire, T^quation (X) sera satisfaite en supposant que ^ satisfait k I'^quation
aux differences partielles (D — ydx) <f> = 0,
EIn posant «
□ = ctn3«^_j + 2a^i3a^ . . . + noida^,
on fera un raisonnement analogue par rapport k Tdquation (F); et il sera ainsi demontre
que <f> doit satisfaire aussi k r^quation k differences partielles (D — «9y)^=0; done
enfin, on a le suivant
TH]£oBi3iE. Tout covariant ^ de la fonction
(ao, Oi, ... an){x, y)~,
satisfait aux deux Equations k differences partielles
(nya,)0=o, (n^y)<^=o, (.i)
oil
«
131] NOUVELLES RECHERCHES 8UR LES OOVARIANTS. 167
et r^proquement toute fonction, homog^ne par rapport aux coefficients et par rapport
aox variables, qui satisfait k ces ^uations, est un covariant de la fonction donn^.
Par exemple, Vinvariant ^^clc — I^ de la fonction cux^ + 2hxy + cy* satisfait aux
Equations
et le catfariant = (ac — 6*)«'+(a9 — 6c)a?y + (63 — c*)y* de la fonction aa:^+Sba^y\3cxi^+dy*
satis&it aux Equations
(aa^ + 2Me + 3cadya,)0 = O, (3d0e + 2ca6+W«ajay)<^ = O.
n est clair qu'en ne consid^rant que les fonctions qui restent les mSmes en prenant
dans un ordre inverse les coefficients Oq, Oi, ... On et les variables x, y, respectivement,
les covariants seront d^finis par Tune ou I'autre des Equations (A), et qu'il n'est plus
n^cessaire de consid^rer les deux Equations. Cela posd, on trouve assez facilement les
comariants par la m^thode des coefficients inddtermin^. Mais il y a & remarquer une
circonstance de la plus grande importance dans cette throne, savoir, que Ton obtient
de cette mani^ un nombre d'^uations plus grand qu'il n'en faut pour determiner
les coefficients dont il s'agit Ces Equations cependant, ^tant li^es entre elles, se r^uisent
au nombre n^cessaire d'^uations ind^pendantes.
Cherchons par exemple pour la fonction a^ + 3ha^y + 3cd7y' + dy* un invariant ^ de
la forme
= ila«* + 5a6cd + Ooc' + Gh'd + D6«c»,
contenant les quatre coefficients ind^termin^s A, B, C, D, £n substituant dans T^quation
(085+ 263c + 3cdd) ^ = 0, on obtient
(3C + 25) a6»d + (3B + 6(7 + 2Z))a6c> + (6^ + 5) ac»d + (3(7 + 42)) 6»c = ;
Of les quatre ^nations donn^es par cette condition, se rdduisent k trois Equations
ind^pendantes, de sorte qu'en fsdsant par exemple 4=— 1, les autres coefficients seront
d^tennmds, et Ton obtient le r&ultat connu :
(^ =  d^« + 6a6cd  4ac»  46»d + Wc\
La circonstance mentionn^ cidessus s'oppose k r&oudre de la mani^re dont il
^^t, le probl^me de trouver le nombre des invariants d'un ordre donn^: probl^me
^ a toujours brav^ mes efforts.
Avant d'entamer la solution des Equations {A)y je vais d^montrer quelques propri^t^s
^^rales des covariants, et des invariants. Pour abr^ger, je me servirai du mot pesanteur,
en disant que les coefficients Oo, Oi, &c., ont respectivement les pesanteurs — ^n, 1 — ^n,
^) que les variables x, y ont respectivement les pesanteurs ^, — ^, et que la pesanteur
168 NOUVELLES RECHEECHES 8UR LES OOVARIANTa [l3l
d'un produit est ^gale k la somme des pesanteurs des facteuis. Cela pos^, je dis que
tout covariant est compost de termes dont chacun k la pesanteur z&o. Pour d^montrer
cela^ j'^cris:
(na0y)(nya,) = nnya,Daj0yn+«ya.ayH«a,;
cela donne
«
or, en faisant attention aux valours de D, D, savoir
gn = (DD) + nooaa^ + 2(n  l)a,a«^ ... + n 1 a^,a«_^,
oil, en formant les produits (DU), (UD)) I^ quantit^s Oo, Oi, ... o^ sont cens^ non
affect^s par les symboles da^,da^,... da^ de la difii^rentiation, on en tire
nnnn = nao3a +(n2)oi9a ... — nonda
=  2 {(0  hn)aoda^ + (1  ^n)a,da^ . . . + (n  in)anda,} =  26,
en reprdsentant par 6 Texpression symbolique entre les crochets. De \k enfin on obtient :
Or en supposant les deux parties de cette ^nation symbolique appliqu^ au covariant
^, la partie gauche de T^uation s'^vanouit en vertu des ^nations (A) et T^uation
se r^uit k
(8 + ia«xiyay)0 = O; (B)
ce qui est une nouvelle Equation k differences partielles, k laquelle satisSut le covariant
<!>. II est ais^ de voir que cette Equation exprime le th^r^me dnoncd cidessus,
savoir que tout covariant est compost de termes de la pesanteur ziro.
n suit de Ik, en considdrant un covariant
<^ = (ilo, Au ... At)(x, yy
qu'un coefficient quelconque At aura la pesanteur i — ^8, ou bien que les pesanteurs
ferment une progression arithm^tique aux diff(^rences 1, et dont les termes extremes
sont — i«, +^s.
Substituons maintenant cette valeur de <f) dans les ^nations (A). La premie
^nation donne d'abord:
nilo = 0, DAi^Aoy DA^^iAu ...nA, = sA^^ (a)
Cela est un syst^me qui ^quivaut aux deux Equations
n'.il, = 0, = y'.e°f.il, (cT)
131] NOUVELLES RECHERCHES 8UR LES CO VARIANTS. 169
De mSme, la seconde ^uation donne
BTSt&me qui ^uivaut aux deux Equations
[!l'+Mo = 0, 4>^afe^».Ao (/SO
On voit que A^ satisfait aux deux ^uations
nA = 0, 6'+^ilo = 0, (7)
et en supposant que cette quantity soit connue, on trouve les autres coefficients
^1, A^ ...,Ag par la seule differentiation, au moyen des ^uations 0^). Or cela dtant,
je dis que les ^nations (a) seront satisfiedtes d'ellesmSmes. "En. effet: des Equations
Dilo=0, [!lulo=«ili on tire 00^=0, DDA^^sDAu et de ]k (DD  nn)ilo = «nili.
Or nous avons d4jk vu que CD — 00 = 26, et TAjuation (B) donne B.Aq+^s.Ao^Oi
done r^uation (DO — OiJ)ilo = — «04.i se r^uit k Ao=QAi: ^nation du systfeme
(a). De la mSme mani^re on obtient les autres ^nations de ce syst^me. On pent
dire que Ton aurait pu determiner ^galement le coefficient Ag au moyen des ^nations
D4. = 0, D'.As = 0, (8)
et de ]k les coefficients il^i, ... Aq par les ^nations (a).
Prenons par exemple un covariant (Ao, Ai, A^) (a?, y)' de la fonction cubique
(w* + 36a^ + 3ca?y* + dy*. A^ doit satisfaire aux deux ^nations
(adt + 2hdc + ^d)Ao = 0, (36a« + 2cdt, + ddcfAo = 0.
Ces Equations sont en effet satisfiedtes en mettant Ao = ac — b^. On a done les Equations
2Ai = {3bda + 2cdb + ddc)Ao, A^ = (Sbda + 2ddt + ddc)Au
pour determiner Ai, A^; ce qui donne 24i = ad — 6c, il, = 6d— c", et on est conduit
unsi au covariant mentionn^ cidessus, savoir k
(ac  b^)a^ + (ad be) ay + (bd  c^) y\
Soit maintenant
ai^da^'Of^^yda^ ... ±r3a.=A,
on aura
nA = (gA), AD=(AD)y.
C. n. 22
170 NODTVELLES RBCHERCHES SUR LES COVARIANTS. [131
oil dans (DA), (AD) les quantity Oq, Oi, ... sont cens^ non affectdes par lea symboles
da,> do,, &C. de la di£fdrentiatioD. Cela donne
DAAD = y^.
Or 3aj A — Adx = ^ , done :
(Dya.)A = A(Dya,),
et de m6me:
(naj0y)A = A(na»yX
Appliquons ces deux Equations symboliques k un covariant 0. Les termes k droite
s'^vanouissent k cause des ^nations (A), et Ton obtient les deux ^nations
. (Dya«)A0=O, (n'/cdy)A4>^0,
o'estji^dire : A0 sera aussi un covariant de la fonction donn^e. Par exemple de Vinva
riant
on tire le covariant
savoir :
( a^ + 3a6c  2i» )«»
3( aM2ac«+ 6«c)aj^
+ 3( acd26«dH 6c«)ay«
 (a(?+3acd2c» )y»;
r&ultat ddjd. connu.
Essay ons maintenant k int^grer les ^nations {A)\ savoir:
(ny9«)0O, (Daj0y)^ = O.
Pour int^grer la premiere, je reviens k une notation dont je me suis ddjij servi dans
ce mdmoire et j'^cris
Oj' ssOi + XOo,
Oa' = a, + 2\ai + X'Oo,
On' = On + w\a»i • • • + X**ao.
131] NOUVELLES RECHERCHES 8UR LES OOVARIANTS. 171
En faisant X = , ce qui donne a^ = 0, on voit sans peine que Ton satisfera ^
r^uation, en mettant pour <^ une fonction quelconque de quantitds Oq^ a^, ... a^^
^ + ^y> y \ 1® nombre de ces quantity ^tant n + 2. Et cela est la solution gdndrale de
r^uation.
Ce r&ultat doit 6tre substitute dans la seconde Equation, savoir dans (U — ^y)<^=0.
Four cela, imaginons que les quantitds Oo, Oi, ... On, ^, y soient exprimdes en fonction
de Oo'f CLtt ..• fltn'i ^> y et Ci ; puisque ^ est fonction des seules quantitds Oq', a^ ... On',
X, y, r^uation r^sultante doit 6tre satisfiedte, quelle que soit la valeur de Oi. Or on
trouve que cette Equation r&ultante a la forme L + Mai=:0: done il faut qu'on ait
k la fois les deux ^nations Z = 0, M=0. (Je renvoie k une note les details de la
r^uction.) En demi^re analyse, et en remettant dans les ^nations X — 0, M = les
quantit^s Oo, a,, ..., On au lieu de Oo^ o^^ ..., an\ je trouve les r^sultats suivants tr^
simples, savoir, en ^crivant
e = (0  ^n)a,da^ + (2  ^n)a,d^^ + (3  in)a,a«^ . . . + (n  in)anda^ :
•D = (n  2)a^da + (n  S)a,da ... +and,
«i"
Les ^nations dont il s'agit sont
{(nl)a,enya,)aoCn^,)}^=0, (C)
(8 + ia«,Jyay)^=0, (D)
et il y a & remarquer qu'on obtient T^quation (C) en ^liminant entre les ^nations
(A) le terme doi^; ^t puis, en mettant ai = 0, on tire I'^uation (D) de T^uation
(B), en y mettant de mSme Oi = 0. II y a & remarquer aussi que la fonction <f> qui
Kitis&it aux ^nations (C, D), est ce que devient un covariant quelconque <^, en y
ntettant Oi^O. On obtient d'abord la valeur g^n^rale en changeant Oo, a,, ... , On en
^'> 0,', ..., On^ et en mettant apr^ pour ces quantit^s leurs valeurs en termes de
^f Oi, 0,, ..., On La solution du probl^me des covariants serait done effectude si Ton
pourrait int^grer les Ajuations (C, D).
Or la quantity Oq entre dans Tdquation ((7) comme constante, et Ton voit sans
peine que cette ^nation pourra 6tre int^grde en mettant Oo = 1 ; puis, en ^rivant dans
le r&ultat — , — , ... — au lieu de Oj, 0^, .., a^ et en multipliant par une puissance
qnelconque de Oq, le r&ultat ainsi obtenu, serait composd de termes de la mdme
t^nteur; et en choisissant convenablement la puissance de Oo, on pourrait faire en
sorte que ces termes fussent de la pesanteur z4to. Mais Tdquation (D) ne fait qu'exprimer
que la fonction ^ est compos^e de termes de la peaanteur z^ro; le rdsultat obtenu de
^ mani^re dont il s'agit, satisfera done par luim6me k Tdquation (D), et il est
pennis de ne fiedre attention qu'& I'^quation (C). Dans la pratique on intdgrera cette
22—2
172 N0UVELLE8 BECHEBCHES 8UR LES GOV ABI ANTS. [131
^nation en ayant soin de faire en sorte que les solutions soient de la peswnJteur z^ro,
ce qui peut Stre effectu^ en multipliant par une puissance convenablement choisie de
Oo. Et puisqu'en faisant abstraction de cette quantity Oo, T^quation {(J) contient n + 1
quantit^s variables, savoir a,, a,, ...,an, x, y, la fonction ^ sera une fonction arbitraire
de n quantity ; et en supposant que cette fonction ne contienne pas les variables
w, y (cas auquel serait ce que deviendrait un invariant quelconque en y mettant Oi = 0),
^ sera une fonction arbitraire de n — 2 quantity.
La mSme chose sera ^videmment vrai, si Ton r^tablit la valeur gdn^rale de a^ :
done tout invariant sera une fonction d'un nombre n — 2 dUnvariantSy que Ton pourra
prendre pour primitifs; et tout covariant sera une fonction de ces invariamts primiti&
de la fonction donn^ (laquelle est dvidemment un de ses propres covariants), et d'un
autre covariant que Ton peut prendre pour primitif. Cela ne prouve nullement (ce qui
est n^nmoins vrai pour les invariants, k ce que je crois) que tout invariant est une
fonction rationnelle et int^grale de n — 2 invariants convenablement choisis, et que tout
covariant est une fonction rationnelle et int^grale (ce qui en effet n*est pas vrai) de
ces invariants, de la fonction donn^e, et d'un covariant convenablement choisL
Le cas n = 2 fait dans cette th^rie une exception. On sait qu'il existe dans ce
cas un invaria/nt, savoir ac^l^ qui, selon la th^rie g^n^rale, ne doit pas exister, et
il n'existe pas de covariant, hormis la fonction donn^e ellemSme. Or cette particularity
peut 6tre aisdment expliqu^.
Le cas 71 = 3 rentre, comme cela doit 6tre, dans la th^rie g^n^rale. En effet, il
existe dans ce cas un invariant, savoir la fonction — a'cP + 6ahcd + 4ac* — 46'd + 36'c*
cidessus trouv^, et tout covariant de la fonction peut 6tre exprim^ par cet invariant
de la fonction donn^ ellemdme, et par le covariant (oc — 6*)«' + (ad — 6c)a;y + (M — c*)y'
cidessus trouvd II en est ainsi par exemple pour le covariant de troisi^me ordre
par rapport aux variables et aux coefficients; car en repr^ntant par ^ le co
variant dont il s'agit, par H le covariant du second ordre, par u la fonction donn^
aa? + Zbah/ + 3ca;y' + dy* et par V Vinvariant, on obtient I'^uation identique
^' + Dw* = — 4jEr". Je £aas mention de cette ^nation, parce que je crois qu'elle n'est
pas g^n^ralement connue.
Je vais donner maintenant quelques exemples des Equations (C et Z>). Soit d'abord
71=3, et supposons que 4> ^^ contienne pas les variables x, y: ^ sera une fonction
de a, c, d, et les Equations reviendront k
{Qd'dd  adde)^ = 0, ( 303^ + 09^ + 3(©d)^ = 0.
Les quantit^s oc*, a*d^, dont chacune est de la pesanteur ziro, satisfont par 1& d. la
seconde Equation, et en mettant ^ = ila'cP + Coc', on obtient 4il — C=0, en vertu de
la premiere Equation; ou en faisant il = — 1, cela donne C= — 4; de Id. on tire
^ = — a*cP — 4ac*, et la solution gdn^rale est $ = jF(— a'cP — 4ac*), F ^tant une fonction
quelconque. La formule plus g^n^rale ^^F{a, ^a^d^—^Kuf) satisferait sans doute k la
131] NOUVELLES RECHERCHES SUR LES CX) VARIANTS. 173
premie ^uation, mais pour que cette valeur satis&sse k la seconde ^uation, il faut
que la quantity a, en tant qu'elle n'est pas 'contenue dans — a^c^ — isoc^, disparaisse.
Ainsi la valeur donn^ cidessus, savoir ^ = jF(— a*(? — 4ac*), est la solution la plus
gt^n^rale des deux ^nations.
Ecrivons a, c , d 1 — r au lieu de a, c, d, et 6 au lieu de $, nous obtenons :
4> = F{'a?d} + 6abcd  4ac»  46W + 36V);
ce qui est Texpression la plus g^n^rale des invariants de la fonction aic*+3&ij^+3ca:^*+y',
et Ton Yoit que tous ces invariants sont fonctions d'une seule quantity que nous avons
prise cidessus pour Vinvariant de la fonction de troisi^me ordre dont il s'agit
Soit encore n = 4, sera une fonction de a, c, (2, e qui satisfait aux Equations
{2acBc + (a«  9c*) da  licdde] ^ = 0,
{2aaa + d3d + 2eae}^=0,
dont la solution gdn^rale est = F{ae + 3c*, axie — acP — c*), F ^tant une fonction quel
conque. On voit par Ul qu'il n'existe que les invariants ind^pendants ew — 4cd + 3c*,
ace + 2icd — ocP — 6*6 — c*. Ce r&ultat est connu depuis longtemps.
Soit enfin n = 5, <^ sera une fonction de a, c, d, e, f qui satisfait aux Equations
{3adde + (2ae  12c*) da + (a/ 16cd) de  20cedf] ^ = 0,
On sait qu'il y en a une solution de quatri^me ordre par rapport aux quantity
a, c, d, e, f\ et en prenant la fonction la plus g^n^rale dont les termes ont la pesanteur
z^ro, on aura:
= ula*/» + Bacdf+ Cac^ + Bad^e + Ec^e + F(M^ :
fonction qui satis&it d'ellemSme k la seconde Equation. En substituant cette valeur
dans la premiere ^nation, on trouvera que les coefficients A, B, &c. doivent satisfaire
i ces sept ^nations :
2£ + 2C40il = 0, 35 + 2) = 0, 3C + 42) = 0, 125 + iE' = 0,
QiF 24Z> + 4^* 32C  205 = 0, 6i^162)=0,  24i^ 16iE'= 0,
<I^ se r^uisent cependcuit (ce que Ton n'aurait pas facilement devin^ par la forme
des ^juations) k cinq Equations ind^pendantes. En faisant done il = 1, on trouve
aia^meiit les autres coefficients B, G, &c. et on obtient ainsi :
^ = a*/» + 4acd/+ 16acc*  12ad*e + 48c»c  32c*d* :
▼aleur qui pent 6tre tirde d'une formule prdsent^e dans mon m^moire sur les h3rper
d^rminants, [161 ^^ y f^^dsant 6 = 0.
174 NOUVELLES BECHERCHES 8UR LES COVARIANTfiL [l31
J'ai donn^ cet exemple pour faire voir qu'il serait impossible de deduire du Dombre
suppose connu des coefficients inddtermin^s qui correspondent, k un ordre donnd, le
nombre des invariants de ce mSme ordre. II est done inutile de pousser plus loin cette
discussion.
Note 1 8ur Vintigration des iqwiUons {A).
En ^rivant comme cidessus:
n = afia, + Soiaao. ... + nani3a,,
n = noxda. + (n  \)a^da, ... + CLn\_^^
il s'agit de trouver une quantity ^, fonction de Oq, Oi, ... an, a? et y qui satisfasse k
la fois aux ^nations
(na?ay)0=o.
Pour int<^grer ces ^nations, j'^ris, comme plus haut :
do' = 00,
a/ = ai + Xoo,
ai^^a^^ 2Xai + Va©,
On' = On + nXa,i_i ... +X*ao,
et aussi of = x — \y, y' = y. Cela pos^, je fais remarquer d'abord que ^r = ao'» ^=2ai',
et ainsi de suite. Eln consid^rant X comme fonction quelconque de Oo, Oi, ... On, et en
supposant que ^ soit une fonction de Oo', Oi^ ... a^, oiy y\ on parvient assez facilement
k r^quation identique (D — y9aj)^ = (l + D^) (III'~y'3«')0> ^^ Q' ©st ce que devient
D, en y ^rivant Oo', Oi', ... a^ au lieu de Oq, Oi, ... On.
Nous pouvons done satisfaire k la premiere ^nation, en determinant \ au moyen
de l + nx=0: &}uation qui serait satisfaite en ^rivant \= — — , ou, si Ton veut, en
determinant X par a^^O. Done, en supposant toujours que X ait cette valeur, 4>
sera une fonction quelconque de Oo', a,', ... a^, of, y\ c'estidire d'un nombre n + 2
de quantit^s. Ce sera done \k (comme on aurait pu facilement prdvoir), la solution
g^n^rale de la premifere Equation. Or en consid^rant <f> comme fonction de Oo', a,', ... a„\
of, yfy ou, si Ton veut, de Oo', a^, a,', ... a^, of, y' (oil Oi' = ai + Xo© = 0), et en
131] NOUVELLES RECHERCHES SUB LES COVAMANTS. 175
sabstituaDLt cette valeur dans T^quation (D — a^y)0 = 0, on voit d'abord que la variation
de la quantity X fonmit au r^sultat le terme
(„a.gH«l)«.£)(n'j^.)*;
et puisque na ^ + (^""^)^^'^ r^uit & n^(n — 1) — ^, ou enfin & V^^ —,
ce terme devient
Le terme —oi'dy,^ se r^duit & — (a?' + XyO(""^«'+3y')^» savoir &
et en mettant pour un moment
M— woi(9fl,' + X9aj' +X*3,^/)
+ (nl)a2( ao,' +nX«^aa;)
+ an( 9a^/+wXa«^,),
nous obtenons
c'estidire
€•0
Or en supposant que D' est ce que devient D en y ^crivant a©, Oi', ... a»' au
lieu de Oo* ai, ... On, et en posant
e' = (0  in)ao'aa.' + (1  in) Oi'ao,^ + . . . (n  iii)an'aa;,
on obtient, aprte avoir fait une r^uction un pen pdnible :
M^ + X«n'<^ = n'0 + 2X8'^,
(en effet les coefficients de da,'<^, 'da{^ &c. aux deux cdtes de cette ^nation deviennent
les mSmes aprte des r^uctions convenablea) Done enfin on a
(D  a0v)* = ([i'  a;^,^)^  ^^^^^^ (D'  y'3^)^ + 2X(e' + J<c^^  iy'8^)^ = 0.
ott bien, puisque cette Equation doit Stre satis&ite inddpendamment de la quantity X (qui
fleule contient Oi), elle se decompose dans les deux ^nations
{aii^  x'Z^)  (n  1) ai{U'  j/a^)} <^ = 0,
176 NOUVELLES RECHERCHES SUB LES COVARIANTS. [131
lesquelles, en y mettant d'abord a^^O, puis en remettant Oo, a,, ..., On, a?, y an lieu de
Oo', cLtf ... , Oni «^» y'l et en ^crivant 0, O, •Q, d au lieu de ^, O, D, D, donuent en
effet les Equations (C, D) dont je me suis servi dans le texte.
Note 2.
Je vais r^sumer dans cette note quelques formules qui feront voir la liaison qui
existe entre les invariaiUs d'une fonction de fii^me oidre et de la fonction de (n — l)i&me
ordre que Ton obtient en r^uisant & z^ro le coefficient de y*^, et en supprimant le
facteur x.
IV convient pour cela de consid^rer une fonction telle que
(Oo, Oi, ... an){^,y)n = a^ + (h^~^y ... +a„y~,
dans laquelle n'entrent plus les coefficients numdriques du bindme (1 + x)\
Ecrivons
{a^,Q^, ... a„)(a?, y)« = ao(a?aiy)(a?a^) ... (a?any);
je t&che d'abord & repr&enter les invariants au moyen des racines ai, a„...,a„, et
j'^tends pour le moment le terme invariant k toute fonction, sym^trique ou non, des
racines qui ait la propri^t^ caract^ristique des invariants: fonctions qui jusqu'ici ont
4t6 consid^r^es tacitement comme rationnelles par rapport aux coefficienta
Mettons d'abord
V = Oo^^ioi  a^fioi «,)»... ((Vi  OnY ;
cette quantity V qui, ^gal^ k ziro, exprime T^galit^ de deux racines, et que je vais
d^rmais nommer le Discriminant de la fonction, sera une fonction rationnelle des
coefficients, et d'un invariant proprement dit. Mais de plus, toute fonction telle que
(Oi — a,)**(ai — a,)*, ..., dans laquelle la somme des indices des facteurs qui contiennent
ttx, celle des indices des facteurs qui contiennent a,, &c. sent ^gales, sera un invariant;
et en r^unissant ces fonctions, pour trouver une somme en fonction 8}anetrique des
racines, on obtiendra des invariants proprement dits. Cela soit dit en passant. Pour
le moment il suffit de prendre les invariants les plus simples, savoir ceux de la forme
(«!  tt») (g  q^)
(ax  a,) (a,  fl^i) '
lesquels en effet sent des rapports anharmoniques de quatre racines, prises k volontd. Soient
Oi» Os, ••.>0»i 1* fonction qui vient d'etre ^crite et les fonctions que Ton en tire en
mettant a^, fl^j, ...,an au lieu de a^. Les fonctions V, Q^, Qj, ...,Q,i_, seront des
invariants ind^pendants, et le nombre de ces invariants est n — 2. Done, tout autre
131] NOUVELLES RECHERCHES SUB LES CO VARIANTS. 177
invariant sera une fonction des quantity V, Q^, Q,, ..., Qn^, Soit maintenant an = 0»
et cEn 1ft racine qui devient dgale k z4ro. Lea qucuitit^ Qi, Qa* •••> Qn^ seront toujours
des rapports anharmoniques de quatre racines de T^uation da (n — l)i6me ordre. II
n'y aura que la seule quantity Qn^ qui change de forme, et elle ne sera pas un
intfariant de la fonction du (n — l)i^me ordre. On voit aussi d'abord que le discriminant
V se r^uit k a*niVo» en exprimant par V© le discriminant de la fonction du (n — l)ifeme
ordre. (Cest je crois M. Joachimstbal qui a le premier remarqud cette circonstance.)
Done, en supposant an = 0, Vinvariant de la fonction du ni^me ordre deviendra une
fonction de a^n^o, Qi, Q,, ... Qn^ et d'une quantity X qui n'est pas un invariant de
la fonction du (n — l)ifeme ordre, mais qui sera toujours la mSme quel que soit Tinvariant
dont il s'agit. En consid^rant les invariants proprement dits de la fonction du (n — l)i^me
ordre, on pent former avec ces irivariants des quotients /i, /„ ...,/,^^ du degr^ z^ro
par rapport aux coefficients. Nous pouvons remplacer par ces quotients les quantit^s
Qii Qt) "iQnrii et dire que Vinvariant de la fonction du ni^me ordre, en mettant an = 0,
deviendra une fonction des quantit^s a'»_iVo, A, /,, ..., /,^^ et X.
Ces thdor&mes auront, je crois, quelque utility pour les recherches ult^rieures: je
les laisse k c6t^ maintenant, et veux presenter une m^thode assez simple pour calculer
les discriminants.
Four cela je remarque que les Equations {A\ en changeant, comme nous venous
de le £Edre, les valours des coefficients, donnent pour les invariants :
(naoda, + (n  l)aida, . . . + a^^ida) = 0,
(aida. + 2a,'aa, • • • + nanda^)4> = ;
et ces ^nations seront satisfiedtes en mettant pour <f> le discrimina/tU V. Or, pour
fln=0, la fonction V devient a*«_iVo, ou, si Ton veut, — a'niV©; done V sera g^nerale
nient de la forme
oil On*^^ est la puissance la plus dlev^ de an Done, en supposant que Vq soit connu,
et en mettant la premiere des Equations Writes cidessus sous la forme (^+a»_i3a^)V=0,
0^ l'=na©9ai + (n — l)oi3fl, ... +2aw»,3a , on obtiendra par la seule diff(^rentiation les
efficients B, C, &c. EIn effet, cette Equation donne
a^iB^F{a*^,V,l %i^,G — F{B\ San^,D = ^ F(C) ;
et ainsi de suite.
En supposant par exemple n = 3, consid^rons la fonction du troisi^me ordre
^^ ixfmminani de okc* + ^xy + yy* sera 4ay — /S*. Nous avons alors
an. 23
178 NOUVELLES BECHERGHES SUB LES CJOVABIANTS. [131
et en mettant ^=3a9^ + 2i98y, B, C seront donn^ par
c'estJidire B ^ ISafiy  *fi*, C = 27a», et de ]k:
V =  27 fl?S« + ISa/SyB  407'  4/9»S + ^Sy :
valeur qui correspond en efifet k la forme ordinaire
V = a«* + 6a6od4ac«4W + 86V,
en changeant d'une mani^re convenable les coefficient&
Londres, SUme Buildings, 23 Fhr. 1852.
132]
179
132.
REPONSE A UNE QUESTION PROPOSEE PAR M. STEINER
(Au%abe 4, Crelle t. xxxi. (1846) p. 90).
[From the Jcumal far die reins und angewandte Mathematik (Crelle), torn. L. (1855),
pp. 277—278.]
En partant des deux th^rfemes :
I. Qu'U existe au moins une surface du second ordre qui touche neuf plans donn&
quelconques ;
n. Que le lieu d'intersection de trois plans rectangles qui touchent une surfisM^e
^^ second ordre est une sphere concentrique avec la surface, tandis que pour le para
^loide cette sphere se r^uit It un plan,
^ Steiner suppose le cas d'un parall^epipkle rectangle, ou mdme d'un cube P
^^ d'un point quelconque D, par lequel passent trois plans rectangles. Les six plans
^^ parall^lepipMe P et les trois plans qui passent par le point D seront touchy d'une
^^'fece F du second ordre (I.), et les huit angles E du parall^epipfede P et le point
^ doivent done se trouver tons les neuf sur la surfiek^e d'une sphere, ou dans un
P^^^ (XL). Les huit angles E sont en efifet situ6s sur la surfiek^e d'une sphere,
^^t^rmin^ par eux ; mais le point D ^tant arbitraire, ce point en g^n^ral ne sera
P^B situ^ sur cette surface sph^rique, de mani^re que les neuf points 8E et D ne
^<t>nt ntuds, ni dans une surface sph^rique, ni dans un plan; ce qui ne s'accorde
P^ avec le th^orfeme II. Cela etant, M. Steiner dit, qu'il y a It prouver que la
^xitradiction n'est qu'apparente, et que tout cela n'affaiblit pas la validity g^n^rale des
deux th^rfemea
II s'agit de savoir ce que devient dans le cas suppose par M. Steiner la surface
du second ordre qui touche les six plans du parall61epipMe P et les trois plans qui
23—2
180 lUiPONSE 1 UNB QUBSnON PROPOSJ^ PAR M. 8TEINER. [132
passent par le point D. Cette surfSeu^ sera en eifet la conique aelon laquelle Vinfini,
consicUrS comme plan, est coupi par un cdne ditermind, pris la position du sommeL
En efifet, menons par un point queloonque de Tespace trois plans parall^les auz plans
du parall^epipMe P, et par le point D trois autres plans parall^les k ces plana Ces
six plans seront touch& (en vertu d'un th^rfeme oonnu) par un cdne ddtermin^ du
second ordre, et on pent dire que ce cdne, quelle que soit la position de son sommet,
rencontre Tinfini, consid^rd comme plan, dans une seule et m£me conique (cela n'est
en effet autre chose que de dire que deux droites parallfeles rencontrent I'infini, con
siddr^ comme plan, dans un seul et m£me point). Le cdne dont U s'agit aura la
propridt^ d'dtre touchy par une infinite de syst^mes de trois plans rectanglea En
effet: le plan passant par le sommet, et perpendiculaire k la droite d'intersection de deux
plans tangents quelconques sera un plan tangent du cdne; les plans d'un tel syst^me
seront aussi des plans tangents de la conique mentionn^e cidessus: done le sommet
du cdne sera le point d'intersection de trois plans rectangles de la conique; et ce
sommet ^tant un point entiferement ind^termin^, le lieu de Tintersection des trois plans
tangents rectangles de la conique, sera de mdme absolument inddtermin^, ou si Ton
veut, ce lieu sera Tespace entier pr&s les points k une distance infinie. La contra
diction apparente dont M. Steiner parle, a par cons^uent son origine dans Tind^ter
mination qui a lieu dans le cas dont il s'agit. Dans tout autre cas, le point
d'intersection des trois plans rectangles de la surface du second ordre est parfidtement
ddtermin^, et les th^rfemes I. et IL sont tons deux l^gitime&
133]
181
133.
SUE UN THEORilME DE M. SCHLAFLL
[From the Journal fQ/r die reine wnd angeivandte MathemaiUe (Crelle), torn. L. (1855),
pp. 278—282.]
On lit dans (§13) d'un m^moire tr^ int^ressant de M. Schlafli intituld "Uber
die Resultante eines Systems mehrerer algebraischer Oleichungen" {M4m. de VAcad, de
Vienne, t TV. [1852]) un tr^ beau th^r^me sur les R^sidtants.
Pour £Eure voir plus clairement en quoi consiste ce th^reme, je prends un cas
particulier. Soit
F=ar» + 2)ftry + 7y" =(«, A y)(x, yy.
Je fiEds p=a?, ? = «yi ^=y'i et je forme les op^rateurs
81= faa + M + iCSc,
lesquels, operant sur U, donnent
L'op^teur
operant sur V, donne
182
SUB UN THltoBilME DE M. SCHLAFU.
[133
Cela ^tant, soit ^==0 le rSstdUmt des ^uations U^O, F=0, c'estiirdire I'^uation
que Ton obtient en ^liminant w, y entre les ^uations Cr«0» F=0, ou autrement dit,
soit ^ le resultant des fonctions U, V. Pour fixer les id^ j'^cris la valeur de ce
r&idtant comme suit:
^s a, Sb, Sc, d
a, db, 3c, d
«, 2)8, 7
«, 2/8, 7
Je suppose que les op^rateurs 9(, S, € op^nt sur le resultant ^, ce qui donne les
fonctions
a*, «*. S0,
ou en ^rivant pour 81, S9, S leurs valeurs :
et en consid^rant ces expressions comme des fonctions de {, 17, (T, j'en forme le
resultant ^, savoir
Ce resultant O contiendra le carr^ de ^ comr/ie /(icteur; c'est ce qui donne, dans le
cas particulier dont il s'agit, le th^or^me de M. SchlaflL
Q^n^ralement, en supposant que Ton ait autant de fonctions U, V, TT, ... que
d'ind^termin^s x, y, ^, .••» on pent supposer que p, q, ... soient des mon6mes aftf^tf^,...
du mSme degr^ X (il n'est pas n^cessaire d'avoir la s^rie entifere de ces mondmes),
et on pent former des op^rateurs 9(, 93, &c. en mdme nombre que celui des mondmes
p, f, ... avec les ind^termin^es {, 17, ... , tels que ces op^rateurs SI, S, ..., operant sur
les fonctions 17, F, IT, ... (chacun sur la fonction It laquelle il appartient), donnent
^(pf + 917 ...)**, ^(/>f + 917 ...)'*', &c.; t, ify &c. ^tant des mondmes de la forme x^tf^sf^
Cela ^tant, soit <f> le r&ultant des fonctions U, V, IT,...; en operant sur ce
resultant ^ avec les op^rateurs 9(, 9,... et en formant ainsi les fonctions S(^, S^, ...,
soit ^ le r&ultant de ces expressions consid^*^ comme des fonctions de {, 17, &c.
^ contiendra une puissance de ^ comme facteur, et en supposant que /i ne soit plus
petit qu'aucun autre des indices /*, /*',...; 7r = /A/i'...; et o=  + —> + ..., I'indice de
cette puissance sera au moins a . Voil^ le th^rfeme g^n^ral de M. SchUlflL
138] 8UR UN THiOR^ME DE M. SCHLAFLI. 188
La demonstration donn^e dans le m^moire cit^ est, on ne pent plu8» simple et
Aigsjite. E!Ue repose d'abord sur un th^r^me connu (d&nontr^ au reste § 6) qui
pout dtre ^nonc^ ainsi; savoir, en supposant que les Equations {7=0, F&=0, ... soient
satisfisdtes, on aura (prte un facteur ind^pendant de {, 17,...) ^
^<l>^t(p^ + qfj...y, 8^ = ^ (pf + 517. ..>*', &C.
Puis, elle est fond^ sur le th^rfeme d^montr^ (§ 12), savoir: le r&ultant des
fonctions
{oh /, /*,... sont des poljnidmes de degrds /*, /*',... en f, 17, &c., et p, g, ..., t, ^, ...
des constantes quelconques) sera, en supposant que /i ne soit plus petit qu'aucun autre
des indices fi, M^•••> et en posant 7r=/i/i'..., tout au plus du degr^  par rapport
aux quantity t, If, &a Voici cette demonstration, qui suppose aussi que le r&ultant
^ soit indecomposable, Supposons que les coefficients de U, F, IT, ... soient assujettis
k la seule condition d'etre tels que le r&ultant ^ soit un infiniment petit du premier
ordre, il sera permis de supposer que tous ces coefficients des ind^termin^ x, y, ...
ne different des valeurs qui satisfont aux ^nations U^O, F=0, TT^O, ... que par des
increments infiniment petits du premier ordre; le resultant if> sera un infiniment petit
da premier ordre, mais toute autre fonction des coefficients, k moins qu'elle ne contienne
nne puissance de ^ comme fiekcteur, aura une valeur finie, et toute fonction des
coefficients infiniment petite de I'ordre k contiendra ^ comme facteur. Dans cette
supposition les Equations S[^ = 0, S^ » 0, &c. deviendront :
oil/ /',..> sont des polyndmes de degr^s /li, fi\ ... dont les coefficients sont des
infiniment petits du premier ordre. En supposant toujours que /* ne soit plus petit
qu'aacun autre des indices /i, /i^... et en posant 7r = fifi\.., o — — + >...> le resultant ^
fAf fit
du syst^me sera tout au plus du degrd — par rapport aux quantit^s finies t, t^ .... Le
d^ par rapport k tous les coefficients est o; le degr^ par rapport aux coefficients
<le /,/',... sera done au moins o ; c'estJrdire, ce resultant sera un infiniment
petit de I'ordre o , ou enfin, ^ contiendra ^^"^'f^ comme facteur. Or les coefficients
^ U, F, Wf... (assujettis k la seule condition cidessus mentionn^e) etant d'ailleurs
^Mtraires, on voit sans peine qu'il est permis de &ire abstraction de la condition, et
que <t contiendra en gdn^ral cette mdme puissance ^^^^'i^ comme facteur; ce qu'il
^agisBait de d^montrer.
184 SUB UN THltoBilME DE M. SCHLAFLL [l33
Rien n'emp^he que ^ ne contienne une plus haute puissance que ^^^^^i^ comme
facteur, ou que ^ ne s'^vanouisse identiquement. On pent mdme assigner de plus
pr^ que Fa (ait M. Schlafli, des cas o{i ^ s'^vanouit identiquement Soient m, m\ m'\ . . .
les degr& de U, V, W,.,. par rapport i, a?, y, 2r,... , p = mmW ...,« = — + —,+ ^, ... ,
771 771 771
if> sera du degr^ ^ par rapport aux coefficients de U. Soient aussi fi^, fi,,, ... les
degr^s de celles des fonctions S(^, 9^, . . . , pour lesquelles les op^rateurs S[, S, . . .
contiennent des diff^rentielles par rapport aux coefficients de U, p = — I — ... : pour
ces fonctions les coefficients seront du degr6 — — 1 par rapport aux coefficients de U ; pour
les autres ils seront du depr^ — . 4> sera done du degr^ (~ — l)p + ^(o — p), = — o — p,
m \fn J m m
par rapport aux coefficients d^ U, et ^r^'^'* sera du degr^ —a — p"^ (a j, c'est
kdiie du degrd * . p par rapport aux coefficients de U, De m^me, en supposant
que les lettres m', p', ... aient rapport It V, &c., Os^'^''* sera du degrd —,. p\ &c.
lit Uf
par rapport aux coefficients de F, &c. Si Tun quelconque des nombres — . — p,
^, . p', &c. est nigatif, et It plus forte raison, si leur somme 8 . — o est negative^
771 Uf Ut
^ doit s'^vanouir identiquement. En particulier, en supposant que le nombre des
fonctions S[^, S^, ... (c'estltdire le nombre des ind^termin^ ^, 17, ...) soit v^ on aura
a>v , et par cette ndson ^ s'dvanouira identiquement si  (o— i/) est n^gatif, c'est
Itdire si v>a. Je ne parlerai pas ici des cas examine par M. Schlaffi, oil 4> con
tient comme facteur une plus haute puissance que ff^r^'i^.
134]
185
134
KEMAEQUES SUE LA NOTATION DES FONCTIONS
ALGEBEIQUES.
[From the Journal fWr die reine und angewandte McUhematik (CSrelle), torn. L. (1855),
pp. 282—285.]
Je me sers de la notation
«', fi", y\
of\ P\ i\
pour repr^nter ce que j'appelle une matrice; savoir un systime de quantit6s rangdes
en forme de carrS, mais d'ailleurs tout It fait indSpendantes (je ne parle pas ici des
matrices rectangulairea). Cette notation me parait tr^s commode pour la th^rie des
^luations UnAiirea; j'^ris par exemple
(f, V» ?, ...) = (
« * fi 9 7 > •••
/»" iO" /'
X«^> y» «> •••)
pour repr^nter le syst^me des ^nations
f = « x^fi y^ry z ... ,
c. n.
24
186
REMABQUES SUB LA NOTATION DE8 FONCTIONS ALG^BRIQUES.
[134
On obtient par Ik T^quation:
(a?, y, z, ...) = (
a, /3 , 7 , ... r^Xfi Vf (T,
a', 13'. y\ ...
Q^'i ^', y, ...
••/»
qui repr^nte le syst^me d'^uations qui donne x, y, z, ... en termes de {,17, (r> *.• >
et on se trouve ainsi conduit k la notation
— 1
« I ^ > 7 > •••
*'f ^'» 7'» •••
,// jcy^ ^//
de la matrice tnt^er«6. Les termes de cette matrice sont des fractions, ayant pour
d^nominateur commun le determinant formd avec les termes de la matrice originale;
les numiriiteurs sont les determinants mineurs form6s avec les termes de cette mSme
matrice en supprimant Tune quelconque des lignes et Tune quelconque des colonnes.
Soit encore
(a?, y, z, ...) = (
tty Oy Cy...
_/ r/ /
tVy " t Cy...
_// r// //
Cvy ^> Cy...
Xa?, y, z, ...),
on pent ^rire:
.) = (
a',
^ , 7 1 •••
/s*. y....
/9". 7"....
a',
a",
Of C f • ..
Of C y . . .
Of V y • . •
A^f Jft Zf ...},
et Ton parvient ainsi k Yid6e d'une matrice compoaie, par ex.
a , )8 , 7 , ...
«'» ^'> y* •••
/»'' iO'' «,/'
« > P , 7 > •••
Cvy ^> Cy...
Cvy ^1 C f • • .
d'f Vf (j'f ...
On voit d'abord que la valeur de cette matrice compost est
(«» fi> 7> •••X^» ^'» ^"* •••)» (*» ^» 7> •••X^> 6'i 6''* •••)> •••
134]
BEICABQUES SUB LA NOTATION DBS FONCTIONS ALGJ^BIQUES.
187
oil (a, fi, 7, ...Xa, o,\ a"i ...) = «« + i8a' + 7a'' + ... . II faut fiaire attention, dans la com
position des matrices, de combiner les lignes de la matrice It gauche avec les cclonnes
de la matrice It droite, pour former les lignes de la matrice compost. II y aurait
bien des choses It dire sur cette th^orie de matrices, laquelle doit, il me semble,
pr^c^er la th^rie de Diterminants,
Une notation semblable pent dtre employee dans la th^rie des fonctions quad
ratiques. En effet, on pent d^noter par
(
a , fi , 7 , ...
*'> ^» Vf •••
M^' /y ^."
Xf. % fXa?, y, ^)
la fonction linwhlvniaire
et de lib par
la fonction qwxdraiiqaj^
(
a, A, ^r, .
A, 6, /, .
5^, /, c,..
X«> y. 8:, ...)»
oa^ + 6y* + c?* + 2/y2r + 25r^a? + 2Aa;y ...
que je repr^nte aussi par
(a, 6, c, .../, g, h, ...X^, y, ^f ...)".
Je remarque qu'en g^n^ral je repr^sente une fonction rationnelle et int^grale,
homog^ne et des degris m, m\ &a, par rapport aux ind^termin^es x, y, &c., a/, y', &c.,
de la mani^re suivante:
(0X^» y,'"T{^y y', ...)"•' ....
Une fonction rationnelle et int^grale, homog^ne et du degr^ m par rapport aux deux
ind^termin^ d?, y sera done repr^nt^e par
( X^, yT'
24—2
188 BEMABQUES SUB LA NOTATION DES FONCTIONS ALQ]£!BBIQUES. [l34
En intFoduisant dans cette notation les coefficients, j'^ris par exemple
(a, 6, c, d$x, yy,
pour repr^nter la fonction
tandis que je me sers de la notation
(a, 6, c, cCJlx, yy,
pour repr^nter la fonction
(M5* + ba^ + cay* + dy*,
et de mSme pour les fonctions d'un degr^ quelconqua tTai trouvd cette distinction
tr^ commode.
[In the foregoing Paper as here printed, except in the expression in the second line of this page,
)( is nsed instead of y{ : it appears by a remark {CreUe, t. u, errata) that the mannsoript had the inter
laced parentheses J^ . Moreover in the manuscript ( ) was osed for a BCatrix, which was thus distingnished
from a Determinant, but in the absence of any real ambiguity, no alteration has been made in this respect.
In the reprint of subsequent papers from Grelle, the arrowhead }( ^^^ 3[ ^ ^^'^ instead of {J) . ]
135]
189
135.
NOTE SUR LES C0VARIANT8 D'UNE FONCTION QUADRATIQUE,
CUBIQUE, OU BIQUADRATIQUE A DEUX INDETERMINEES.
[From the Journal filr die reine und angewandte Mathematik (Crelle), torn. L. (1855),
pp. 285—287.]
La th^rie d'une fonction k deux ind^termin^es d'un degr6 quelconque, par example
( X^, yn
depend du syst^me des cavaricmta de la fonction, lequel est cens^ contenir la fonction
ellemSme.
Pour une fonction quadratiqiie le syst^me de covariants est
(a, 6, cXa?, yA
ac — 6*.
Pour la fonction cubiqtie, le systfeme est
(a, 6, c, dX^f, y)»,
(ac — h^, ad — bc, hd — (^){x, y)",
( a« + 3a6c  26", aM+2ac*6^, ood26« + 6c», ad>3&cd + 2c»)(a?, y)\
 a*cP + 6a6cd  4ac«  46»d + 36»(J«,
lonctions lesquelles, en supposant qu'on les reprdsente par U, J7, ^, Q, satisfont iden
^^ement It I'&iuation
190
NOTE SUB LES COVAMANTS DUNE FONCTION QUADRATIQUE,
[135
Pour la fonction biqiuidratique, le syst^me est
(a, 6, c, d, cX«?, y)*,
(acb', 2ad2bc, ae + 2bd3(f, 2662cd, ced?Xx, y)*,
ace + 26cd — ocP — l^e — c*,
/^  a»d + Sabc  26»,
a»c 2a6d +900" 66*c,
5a66 +15acd106^,
+ lOa'd  106>6, I (a?, y)»,
+ 6ad6 + 106(?  155ce,
+ a6» +26<fo 9c»c +6c(P,
^ +6e* Sccfe +2d»,
et ces fonctions, en supposant qu'on les repr^nte par U, /, J7, /, O, satdsfont iden
tiquement It I'^quation
J'ajoute k ce systfeme la fonction
/»  27/» = aV  1 2a»6(fo»
+ 54a6*c6*  6a6»d«6
54ac»d« 276V
+ 366»C\P,
ISaVc* +54a%eP6 27a^
180a6c'c26 + lOSabcd^ + 81ac%
646»d« + lOSb'cde  646Vc
qui est le discrimincmt de la fonction biquadratique.
Pour donner une application de ces formules, soit proposd de r&oudre une ^nation
quadratique, cubique ou biquadratique, ou autrement dit : de trouver un facteur liniaire
de la fonction quadratique, cubique, ou biquadratique.
U est assez singulier que pour la fonction quadratique la solution est en quelque
sorte plus compliqu^e que pour les deux autrea En effet, il n'existe pas de solution
sym^trique, It moins qu'on n'introduise des quantit^s arbitraires et superflues; savoir,
on trouve pour facteur lin^aire de (a, 6, cX^, y)* Texpression
(a, 6, cXa, iSXa?, y) + V D .(i8«ay),
oil (a, 6, cX«, /8Xa?, y) denote aaa + b (ay + fix) + c/Sy.
Pour la fonction cubiqrie, T^quation O* + D iT'* = — 4tH* fisdt voir que les deux fonc
tions 4> + I7V — D , ^ — ^V — D soiit Tune et Tautre des cubes parfedts. L'expression
V {H^ + u/  o)}  y [H^  u^  m
135] CUBIQUE, OU BIQUADRATIQUE A DEUX IND^TERMHrflES. 191
sera done une fonction lin^aire de x, y; et puisque cette fonction s'dvanouit pour
U=0, elle ne sera autre chose que Tun des facteurs lin&dres de (a, b, c, d)(x, yy.
Pour la fonction biquadrabiqtie, en partant de T^quation
j'^cris
et je mets T^uation sous la forme
(1, 0,Jf, M)(IH, JUy=^iP^.
Done, en supposant que tJi, vr^, «j, soient les racines de T^quation
(1, 0,  M, MXm, If = 0,
OU plus simplement de T^uation
t!r»Jlf(t!rl) = 0,
ces expressions IH — vtiJU, IH — vrJU^ IH — vr^JU seront toutes trois des earr& de
fonctions quadratiquea L'expression
sera done une fonction quadratique, et on voit sans peine qu'elle sera le carr^ d'une
fonction liniaire. Or cette expression s'^vanouit pour {7=0; done ce sera pr^eis^ment
le earrd de I'un quelconque des facteurs lin^aires de (a, 6, c, d, e){x, yf.
L'^uation identique pour les covariants d'une fonction biquadratique donne lieu
aussi (remarque que je dois k M. Hermite) It une transformation tr^s ^l^gante de
tinUgrdU dliptique da?^V(a, 6, c, d, e^x, 1)*.
192
[136
136.
SUE LA TKANSFOKMATION D'UNE FONCTION QUADRATIQUE
EN ELLEM^ME PAR DE8 SUBSTITUTIONS UNfiAIRES.
[From the Journal fWr die reine und angewandte Mathemoitik (Crelle), torn. L. (1855),
pp. 288—299.]
Il s'agit de trouver les transformations lin^aires d'une fonction quadratique
( )(«! y, ^» •••)* ^ eUemSme, c'estltdire de trouver pour («, y, z, ...) des fonctions
lin^aires de x, y, z, ... telles que
En repr^ntant la fonction quadratique par
(0X^» y» ^^ •••)* =
o.
A.
9* .••
h.
b,
J 9 •••
9.
•
•
•
/.
c , • • •
Xa?, y, er, ...)«,
la solution qu'a donn^e M. Hermite de ce probl^me peut dtre rdsum^ dans la seule
&uation
(x, y, z,...) =
(
a, h,g, ...
K b,f, ...
fft Jf ^f •••
— 1
a, hp, g + fi, ...
h + v, 6, /— ^ •••
o, h + v, g — fi, ...
h — p, b, f+\ ...
— 1
a, h,g, ...
A, 6,/, ...
gf j$ ^f • • •
)\*^ty»^$»")»
oil X, /i, I', ... sont des quantit^s quelconques.
136] SUB LA TRANSFOBMATION d'uNE FONCTION QUADRATIQUE &C. 193
En effet, pour ddmontrer que cela est une solution, on n'a qu'^ reproduire dans
un ordre inverse le proc^d de M. Hermite. En introduisant les quantity auxiliaires
(i> V» K* •••)> ^^ P^^^ remplacer T^uation par les deux Equations
(
(
a, A, g, ...
A, 6, /, ...
9* ji c, ...
a, A, ^r, . . .
K ft, /, ...
5^1 ji c, ...
Xa?, y, z, ...) = (
A^> y» ^> ••./ — \
a, A + i/, g — fi,...
A — v, 6, y + x, ...
*
a. A I/, ^r + A*, ...
A + ^'i &> / — X, . . .
Xf» ^» ?» •••)
Xf» ^» ?» •••)
qui donnent tout de suite d'abord
et puis
a: + x = 2f, y + y = 2i;, ^ + z = 2j; &c.
On obtient par Ul:
(OXx, y, z, ...)» = ( 0X2fx, 2i;y, 2? z, ...)»,
=4(oxf. ^. ?, ...)»4(oxf, ^> r....x^, y. ^^ ...)
c'estildire T^uation
( Xx> y. z, ...)»=( X^'» y. ^^ •••)"»
qu'il s'agissait de verifier.
Je remarque que la transformation est toujours propre. En effet, le determinant
de transformation est
a, h, g ...
A, b, f ...
9,f, c ...
—I
h+v, b, y— X ...
gfi,f+\ c ...
Oi h + v, g — fi ...
h — v, b, y + X ...
fl^ + A*, /X, c ...
— 1
a, A, ^r ...
A, 6, / ...
g> f, ...
Or les determinants qui entrent dans les deux termes moyens, ne contiennent Tun
ou Tautre que les puissances paires de X, fi, v, ... . Done ces deux determinants sont
^ux, et les quatre termes du produit sont rdciproques deux k deux; le determinant
de transformation est done + 1, et la transformation est propra
Pour obtenir une transformation improprey il faut consid^rer une fonction quadratique
qui contient outre les indeterminees x, y, z, ... une indeterminee 0, et puis r^duire k
C. II. 25
194
SUB LA TRANSFORMATION d'uNE FONCTION QUADRATIQUB
[136
z^ro les coefficients de tous les termea dans lesquels entre cette ind^termin^ 0. Les
valeurs de z, y, z, ... ne conidendront pas 0, et en repr^ntant par % rinddtermin^
que Ton doit ajouter k la suite z, y, z, ... , la valeur de % sera, comme on voit sans peine,
^ = — 0; le determinant de transformation pour la forme auz inddtermin^ w, y, z, ,..y0
sera + 1 et ce determinant sera le produit du determinant de transformation pour la
forme aux ind^terminees x, y, z,.., multipli^ par —1. Le determinant de transformation
pour la forme aux indeterminees x, y, z, ... sera done —1, c'estjldire, la transformation
sera impropre.
Au lieu de la formule de transformation cidessus, on peut se servir des formules
(f V> ?, ••) = (
h^p, b , y+x,...
> •••
— 1
a, A, jr,...
9* J* c,...
X*> y» *> •••)»
x = 2fa?, y = 2i;y, z=2f2r, ....
Par exemple, en supposant que la forme k transformer soit
on aura
(f % t •••) = (
a, I/, A*,...
— I/, 6, X, . . .
M> — ' A», c, . . .
•%ax, by, cz, ...),
x«2fa:, y = 2tyy, z = 2fr, Ac,
de mani^re qu'en posant
a, v, ^,...
■ i/, 6, \, . . .
M> X, c, . . .
=*,
oil aura
(x, y, z •••)— f
a, V,
/*«•••
.1
.'. 6,
A>) ...
•
•
•
C • • .
1
it
a
2£
f
2A'
2^
//
25'
6'
25"
2C
2C
2(7"...
Xcue, by, cz, ...),
136] EN ellem£mb par des substitutions lin^ires. 195
ce qui est I'^uation pour la transformation propre en ellem^me, de la fonetion
a^ + &y*f cei* + &K^ On en d^uira, comme dans le cas gdndral, la formule pour la
transformation impropre. On trouvera des observations sur eette formule dans le
m^moire "Becherches ult^rieures sur les determinants gauches" [137].
Je reviens k T^uation gdn^rale
( Xx, y, z, ...)» = ( x^. y. ^. •••)*.
et je suppose seulement que x, y, z, ... soient des fonctions lin&dres de x, y, z, ... qui
satisfont k cette ^nation sans supposer rien davantage par rapport k la forme de
la solution. Cela ^tant, je forme les fonctions lindaires z — sx, y — sy, z — sz, &c., oil s
est una quantity quelconque, et je consid^re la fonetion
(OXx«a:, y«y, z«^ ...)»,
laquelle, en la d^veloppant, devient
(i + «*XO)(^, y, z...y^2s(<>){x, y, Z...XI % r...);
et en d^veloppant de la mSme mani^re la fonetion quadratique
(0)(xar, yjy, z^, ...j ,
on obtient I'^uation identique
/ 1 1 1 \«
( OXx«a?, y«y, z«^, ...)» = «».( 0)(xa?, yy, ^"g^'"')'
Soit n 1^ determinant formd avee les coefficients de fonctions lin^ires x — sx,
ysy, z— «z, &c. En supposant que le nombre des inddtermin^es x, y, z, &c., est n,
sera ^videmment une fonetion rationnelle et int^grale du degrd n par rapport k 8.
Soit de mSme Q' 1^ determinant form^ avec les coefficients de
1 1 1 .
xir, yy, z^> &c.;
r^uation qui vient d'etre trouv^e, donne □« = «** □'«, c'estkdire Q = ± «* D' Cela
&it voir que les coefficients du premier et du dernier terme, du second et de Tavant
dernier terme, &c., sont dgaux, aux signes pr&& De plus, le coefficient de la plus
haute puissance «* est toujours ± 1» et on voit sans peine qu'en supposant d'abord
que n soit impair^ on a pour la transformation propre:
n=(i, p,...p, ix«. ir
et pour la transformation impropre
n = (l, P....P, lX^> If:
Equation qui pent 6tre chang^ en celleci: n = — (1> P, ... P, 1X*> I)** P^^» ^^
supposant que n soit pair, on a pour la transformation propre:
n=(i, P, ...p, ix«, 1)^
et pour la transformation impropre:
n=(i, P....P, ix«» m
25—2
i
196 SUR LA TRANSFORMATION d'uNE FONCTION QUADRATIQUE [136
le coefficient moyen dtant dans ce cas 4geA k z6ro, Ces th^rfemes pour la forme du
determinant des fonctions lin^aires x — «r, y — «y, z — «z,' . . . sent dus k M. Hermite.
II y a ^ remarquer que la forme ( X^> y» z ...y est tout k Mi inddterminfe;
c'est^dire, on suppose seulement que x, y, z, ... soient des fonctions lin^aires de
^> y» ^, •••> telles qu'il y ait une forme quadratique ( X^» y> ^> •••)* P<>^r laquelle
r^uation ( X^, y, z ...)^ = ( X^> y, ^ ^Y ^t satisfaite.
Je regarde d'un autre point de vue ce probl^me de la transformation en elle
mSme, d'une fonction quadratique par des substitutions lin^ires. Je suppose que
X, y, z, &c. soient des fonctions lin^aires donn^es de x, y, z, ... , et je cherche une
fonction lin^aire de x, y, z, &c qui, par la substitution de x, y, z, &c. au lieu de
X, y, z, &c. se'transforme en ellemSme k un facteur prfea Soit (f, m, w, ...X^, y, z, ...),
cette fonction lin^ire, il faut que (f, m, 7i, ...Xx, y, z, ...) soit identiquement =«.(f, m, w, ...)
(a?, y, z, ...), ou, ce qui est la mfime chose, que (f, m, n, ...Xx — «p, y — «y, z — «^, ...) soit = ;
c'estjidire, les quantity Z, m, n, ... seront ddtermin^es par autant d'^uations lin^aires
dont les coefficients sont pr^is^ment ceux de x — «a?, y — sy, z — sz, &c. ; done s sera
d^termind si Ton rend dgal i zdro le determinant formd avec ces coefficients, et Z, m, n, &c.
86 trouveront donnas rationnellement en termes de 8. Cela ^tant, je suppose que les
racines de T^quation en 8 soient a, 6, c, . . . , et ces diffdrentes racines correspondront
aux fonctions lin&ires x^, x^, x^., ... qui ont la propri^te dont il s'agit. Soit ( X^» V* ^* •••)'
une fonction quadratique qui se transforme en ellem^me par la substitution de x, y, z, &c.
au lieu de a?, y, z, &c. Cette fonction pent 6tre exprim^e en fonction quadratique de
Xa, Xft, x«, &C.; quantitds qui, en substituant x, y, z, &c. au lieu de x, y, z, ... deviennent
(t^H y 0X5 ) CX^ , . • » .
Je prends les cas d'une fonction binairef temairej &c., et d'abord le cas d'une
fonction binaire.
En ^crivant d*abord ( X^> yY^i^t ^» C^X^a> x^)*, on doit obtenir identique
ment (A, B, C) (ax«, 6xft)« = (il, B, C)(Xa, x^y, c'estidire ^(a«l) = 0, 5(a6l) = 0,
(7(6*— 1) = 0. Or la solution A=B = C = ne signifiant rien, on ne pent satisfaire k
ces Equations sans supposer des relations entre les quantit^s a, 6; et pour obtenir une
solution dans laquelle la fonction quadratique ne se rdduit pas k un carre, il £Etut
supposer, ou aft — 1 = 0, ou a' — 1 = et 6" — 1 = 0. Le premier cas est celui de la
transforination propre. II donne
a6 = l, ( 0X^» y)» = ZxaX6.
Le second cas est celui de la transformation impropre. II donne
a = + l, 6 = l, ( OX^* y)' = ^VlwXft«.
En passant au cas dune fonction temaire, soit
( X^, y» ^y^(^. B, C, P, (?, jyXxa, Xft, x,)«i
on doit avoir identiquement
{A, B, C, F, G, HXaXa, 6x,, cx,)« = (il, «, C, F, (?, ^x„, x,, x,)«.
136] EN ELLEmMe par DES substitutions LINilAIRES. 197
c'estil.dire^(a«l) = 0, £(6>l) = 0, C(c»1)=0, J^(6c1) = 0, (?(cal) = 0, Hiahl)
= 0, et on voit que pour obtenir une solution dans laquelle la fonction quadratique ne
sse r^uit pas k un carrd» ou k une fonction de deux ind^termin^es, il faut supposer
par exemple a* — 1 = 0, 6c — 1 = 0. On a done dans le cas d'une fonction temaire :
a'=l, 6c = 1, ( X^> y» '2:)» = Zxa' + mx6Xe.
Xa transformation sera propre, ou impropre, selon que a = + l ou a = — 1.
Dans le cas d'une fonction qtuitemaire, on obtient pour la transformation propre:
oi = cd = 1, (<>)(a!y y, z, wy = l XaXb + m XcX^,
est pour la transformation impropre:
a = + l, 6 = l, crf = l, ( X^> y. ^^ ^)' = ^ V + mxfr« + nXcXd.
Dans le cas d'une fonction quinaire on obtient
a*=l, 6c = dc=l, ( X^> y» ^» ^> t)' = Zx«« + mxftXc + wxrfX«
est la transformation est propre ou impropre, selon que a = + 1 ou a = — 1 ; et ainsi
de suite.
Cette mtStbode a des difScultes dans le cas oil T^uation en ^ a des racines
egales. Je n'entre pas ici dans ce sujet.
Dans les formules qu'on vient de trouver, on pent consid^rer les coefficients
£, m, isQ, coqime des quantit^s arbitraires. Mais en supposant que la fonction quadra
t.ique soit donnie, ces coefficients deviennent dAerminis. On les trouvera par la formule
»uivante que je ne m'arrSte pas k d^montrer.
Soient a, /S, 7, &c. les coefficients de la fonction lin^ire x^, a', P\ 7', &c. les
ooefficients de la fonction lin^aire z*, et ainsi de suite; alors, dans les diffi^rentes
fiormules qui viennent d'etre donnas, le coefficient d'un terme Xa' k droite sera
it
et le coefficient d'un terme XaX^ k gauche sera
(tX«» P* 7. •••X<»'> ^> 7'> •••)'
oi!i k denote le discriminant de la fonction quadratique k gauche, et ou les coefficients
des fonctions quadratiques des d^nominateurs sont les coefficients inverses de cette mSme
foiiction quadratique k gauche ^
' Je profite de oette oooaaion pour remarqner oonoemftnt oes reoherches que les fonnules donn^es dans
^ note stir lee foxictioiiB dn seoond ordre (t zumi. [1S4S] p. 105) [71] poor les cas de trois et de qoatre
^^tennin^es, sont ezaolea, mais que je m'^tais iromp6 dans la fonne g^n^rale dn thtor&me. [This correction
U indicated toI. x. p. 589.]
198 SUR LA TRANSFORMATION d'UNE FONCTION QUADRATIQUE [136
L'application de la mdthode k la forme binaire (a, b, c){m, yY donne lieu aux
d^veloppements suivants.
J*&;ris x=aa? + /8y, y=7« + Sy, et je repr^nte par (I, m)(x, y) ime fonetion
lin^aire qui par cette substitution est transform^e en ellemSme, au &cteur 8 pres.
Nous aurons done
{I, mXax + py, ya + Sy)^8 (I, m){x, y) ;
r^uation pour 8 sera
«»«(S + a) + aS)87 = 0;
laquelle pent aussi 6tre ^rite comme suit:
(1, Sa, aS/37X«, 1)' = 0.
Soient 8\ tt' les racines de cette ^nation. (II est k peine n^cessaire de remarquer
que «', 8"y et plus bas P, Q, sont ici ce que dans les formules gdn^rales j'ai repr^
sentd par a, 6 et Xa, x^. De mSme les Aquations p = «'«", p^8'\ p^8"\ obtenues
apr^y correspondent aux ^nations a& = l, a^ = l, &* = !.) On aura
«' + «" =  So, «V' = aSi87,
et les coefficients 2, m seront ddtennin^ rationnellement par s,
Mais on pent aussi determiner ces coefficients par I'^uation
I : m = ia + m7 : 1/3 +mB,
qui pent etre ^crite sous la forme
(/3, Sa, .7X/, m)« = 0.
et en ^liminant entre cette ^nation et Tdquation lx + my = les quantity I, m, on
voit que les fonctions lin^aires Ix + my sont les facteurs de la fonetion quadratique
(/3, S — a, — 7)(y, — a?)", ou, ce qui est la m^me chose, de la fonetion quadratique
je repr^nte ces facteurs par P, Q et je remarque encore que T^uation en s aura
des racines ^gales si
(Sa)» + 4)87 = 0,
et que dans ce cas, et exdttsivement dans ce cas, les fonctions P, Q ne forment qu'une
seule et mSme fonetion lin^aire.
Je suppose maintenant que la fonetion (a, b, c)(x, yY se transforme en ellemSme
par la substitution aa{fiy, yx^By au lieu de a?, y, ou, ce qui est ici plus commode,
je suppose que les deux fonctions sont ^gales k un &cteur prte, et j'^ris
(a, 5, cXouv + fiy, 7a? +8y)» = p (a, 6, c)(x, yy.
136] EN ELLEM^E PAB DES SUBSTITUTIONS LIN]6 AIRES. 199
En d^veloppant cette Equation, on obtient
a;» (a, 6, cXq?  p, 2a7, 7» )^
+ 2 a:y (a, 6, cXa/3, aS + fiyp, yS )^=0.
+ y*(a, 6, cXy8«, 2/3S, S«p),
VoiUl trois ^nations lin^aires pour determiner par les quantitds a, /3, 7, S, consid^r^es
comme donn^, les coefficients (a, b, c) de la fonction quadratique. Les coefficients de
ces Equations lin^aires sont
a«p, 207, 7»,
flf/9, aS + fiy p, 78,
i8», 2/38, 8«p.
Le syst^me inverse par lequel on trouve les valours de a, 6, c, est
«»(aS/87)(aS + i97 + 8")plp»,  jSS (aS  /87) + fl^p,
 27* (oS  /87) + 2a7p. a«8«  )8V  (S* + «») p + p*,
7* (aS — ^87) + 7"p, — 07 (aS — /97) + 7Sp,
fi^(ab^fiy)^/3»p,
 2a^ (oS  /97) + 2i8Sp,
a«(flfSiS7)(aS + /87 + a«)p + p»,
et le determinant, dgal^ k z6ro, donne
(aS)87p){(aS/97 + p)«p(a + S)»}=0:
^uation dont les racines sont
p = aS/97, p={i(a + S)±iV(aS)> + 4)87}>.
En comparant ces valours avec cellos de s', 8'\ on voit que les racines de T^uatiou
en p sont
P = «V', p = «'S p = A
et nous aliens voir que ces valours de p donneut en gdndral les valours F(l^ P*, Q*,
pour la fonction quadratique.
Soit d'abord p = aS — )87 (= ^V), et posons pour abr^ger oS — )87 — p = ^, le systfeme
inverse devient:
(S'p)*/8p. 27, /3S^i9p(Sa), ^<l> + /3p,2fi,
2780p(Sa)27, (a8 + ^yp)<^ p(Sa)»,  2a^<^ + p (S  a) 2/S,
7»^ + 7p.27, a7<^ + 7p(Sa), (a»p)^7p . 2/8,
et en mettant ^==0, les termes de chaque ligne (en omettant un fisK^teur) deviennent
7, ^ (5 — a), fi. On obtient ainsi dans ce cas, pour la fonction quadratique (a, b, c){x, yY
la valour
(% s«, 0X^,yy>
qui est en effet le produit PQ des fonctions lin^aires.
n y a ^ remarquer qu'en supposant (S — a)* + 4^87 = 0, ce qui est le cas pour lequel
p sera une racine triple, il n'y aura pas de changement k hire dans ce r&ultat. La
fonction quadratique est, comme auparavant, le produit PQ des fonctions lin^aires;
200 SUR LA TRANSFORMATION d'UNE FONCTION QUADRATIQUE [136
seulement ces deux fonctions lin^ires dans le cas actuel sont identiques, de mani^re
que la fonction quadratique se r^uit k P*.
Soit ensuite
p = {i(a + S)±iV(a«)» + 4i87}H=«'* ou «"»);
en ^rivant p = «■ et en mettant pour abrdger a8 — ^87 — « (S + a) + «• = x» 1® systfeme
inverse devient
 2iyBx  27J? (S  8) (S + a), {«» + « (8 + a) + aS + ffy} x + 2l3y8 (8 + a),
7^ + 7"« (8 + a),  OTx  7« («  «) (8 + a),
/8»x + /8»« (S + a),
Done, en ^rivant x = ^ ^^ ^^ omettant le fisK^teur 8{S + a), le syst^me inverse devient
et les quantity dans chaque ligne sont dans le rapport l^ : hn : m*, de manifere que la
fonction quadratique est dans ee cas dgale k P^ on Q^. Cela suppose que S + s ne soit
^gal k zdro. En faisant pour le moment p = 1, on en tire la conclusion qu'd. moins
de supposer 8 + a = 0, il n'existe pas de fonction quadratique binaire proprement dite
(fonction non carr^e) qui par la substitution impropre ax^fiy, 7^ + Sy pour x, y, se
transforme en ellemSme. L'^uation Sf a = donne p = aS — /97, qui est une racine
double de Tdquation cubique. On remarquera en passant par rapport k la signification
de r^uation S + a = 0, que Ton a en gdndral:
(a, /8Xap + i9y, yx + Sy) : (7, SXflw? + i9y, ya^ + By)
= (a« + i97)^ + )8(S + a)y : y(Bh a)x{(S' + I3y)y,
et de Ik, qu'en supposant 8 + a = 0, on a
(a, fiXaxVfiy, yx + Sy) : (7, BX^x^fiy, yx+Sy) = x, y.
Cela revient k dire qu'en faisant deux fois la substitution ax + /3y, yx + Sy au lieu de
X, y, on retrouve les quantity x, y, ou que la substitution est pSriodique du second
ordre. II y a aussi k remarquer que dans le cas dont il s'agit, savoir pour S + a = 0,
on a 8" = —8\ et que les deux fonctions lin^aires P, Q restent parfaitement dc^termindes.
Nous venons de voir qu'il n'existe pas de transformation impropre d'une fonction
quadratique binaire proprement dite, k moins que S + a ne soit pas = 0. Mais en
supposant S + a = 0, on voit que les coefficients des ^nations pour a, b, c deviennent
fir 7(««X 7'>
a/9, a(Sa), a7,
/8«, i9(8a), fiy,
136] EN ELLEM^ME PAR DES SUBSTITUTIONS UN^IAIRES. 201
c*e8td.dire : les coefficients de chaque Equation sout dans le rapport de
/8, 8  a,  7,
de manifere qu'en supposant que les coefficients a, b, c satisfont k la seule Equation
(a, 6, cXA Sa, 7) = 0,
oil 0, i9, 7, 8 sont des quantity quelconques, telles que S + a = 0, on aura
(a, 5, cXflw? + i9y, yx + Byy = (QB fiy)(a, h,c){x, y)».
Ce n'est \k qu'un cas particulier de T^uation identique
(a, 6, cXcuc + ^y, yic + Syy + (aSfiy).{a, b, c){x, y)» =
(S + a).(aa + 67, 6(S + a), 6/9 + c8) (a?, y)« +08, Sa, 7Xa, 6, c).(/8, Sa, 7Xy, x)\
II &ut remarquer qu'en supposant toujours T^quation
(a, 6, cX)8, Sa, 7) = 0,
la fonction quadratique (a, 6, oX^> y)^ ^ supposant qu'elle se riduise d, un carri, est
comme auparavant P* ou Q*, c'estildire le carrd de Tune des fonctions lin&ires.
En effet: en ^rivant (a, 6, c)(Xy yY =^ (Ix + myY, T^uation entre i, m serait ^videm
ment (/8, S — a, — 7X^» m)" = 0, de mani^re que I, m auraient les m^mes valeurs qu'au
paravant. J'ajoute que tout ce qui pr^Me par rapport k T^quation
(a. by c){(xx + /9y, 7a? + Byy = p (a, 5, c^x, yY
fait voir qu'^ moins que la fonction quadratique ne soit un carrel, on aura toujours
p = ± (aS — ^97) ; ce qu'on savait d4jk d^s le commencement, et ce qui pent Stre ddmontrd
comme a Toidinaire, en ^galant les discriminants (etc — 6") (aS — /97)" et (ac — 6")p' des
deux cdt^ Je fais enfin p = 1, ce qui donne Tdquation
(a, 6, cXaa? + fiy, 7a? + 8y)« = (a, 6, c^x, yY,
et (en faisant abstraction du cas oil la fonction quadratique est un carr^) je tire de
ce qui prdcMe les r&ultats connus, savoir, que Ton a:
1. Pour la transformation propre:
aS)87 = l,
a:26:c = 7:8 — a:— /8.
2. Pour la transformation impropre:
a/8 + 6(Sa)C7 = 0.
Je crois que cette discussion a iti utile pour completer la th^orie alg^brique de la
forme binaire (a, 5, c){x, y)".
c. n.
26
202
[137
137.
RECHERCHES ULTERIEURES SUR LES DETERMINANTS
GAUCHES.
[From the Journal filr die reine und angewandte Mathematik (Crelle), torn. L. (1855),
pp. 299 — 313: Continuation of the Memoir t. XXXIL (1846) and t xxxviii. (1849);
5S and 69.]
J'ai d6jk donn^ une formule pour le d^veloppement d'un cUterminant gauche.
En prenant, pour fixer les id^es, un cas particulier, soit
12345 12345 =
11, 12, 13, 14, 15
21, 22, 23, 24, 25
31, 32, 33, 34, 35
41, 42, 43. 44, 45
51, 52, 53, 54, 55
(oii 12 = — 21, &c., tandis que les quantity 11, 22, Sec ne s'^vanoiiiasent pas),
formule peut dtre ^crite comme suit:
Cette
12345 12346= 11 .
22 . 33 .
44 . 55
+ 11
. 22 . 33 . <
:45)«
+ 11
. 22 . 44 . (
[35)»
+ 11 .
, 22 . 55 . (
[34)'
+ 11
, oS . 44 . 1
[25y
+ 11
. 33 . 55 . (
[24)'
+ 11 ,
. 44 . 55 . (
[isy
+ 22 .
33 . 44 . (
;i5)«
+ 22 ,
, 33 . 55 . <
[uy
+ 22
. 44 . 55 . 1
(13)*
+ 33 ,
. 44 . 55 . (
[uy
+ 11
. (2345)*
+ 22
. (1346)'
+ 33
. (1245)«
+ 44 ,
, (1235)«
+ 55 .
, (1234)'.
137]
RECHEBCHBS ULTJ&RIEURES BUR LES DJ^ERMINANTS QAUCHES. 203
Lies expressions 12, 1234, <&c k droite sont ici des Pfaffians. On a
12 = 12,
1234 = 12.34 + 13.42 + 14.23
et en Aaivant encore un terme, pour mieux pr^enter la loi:
123456= 12.34.56 + 13.45.62 + 14.56.23 + 15.62.34 + 16.23.45
+12.35.64 + 13.46.25 + 14.52.36 + 15.63.42 + 16.24.53
+ 12.36.45 + 13.42.56 + 14.53.62 + 15.64.23 + 16.25.34.
J'ai trouvd r^mment une formule analogue pour le ddveloppement d'un dAer
fninant gauche bord^, tel que
cette formule est:
al234 /31234 =
; «/8,
«1.
a2,
a3.
a4
lA
11,
12.
13.
14
2/8,
21,
22,
23.
24
3A
31,
82,
33,
34
4/8,
41,
. 11
42,
. 22
43,
. 33
44
al234 /31234
= «/9
. 4
+ ay8 . 12 . 12 . 33 . 44
+ a/3 . 13 . 13 . 22 . 44
+ 0/8 . 14 . 14 . 22 . 33
+ 0/8 . 23 . 23 . 11 . 44
+ 0/8 . 24 . 24 . 11 . 33
+ 0/8 . 34 . 34 . 11 . 22
+ 0/8 . 1234 . 1234
+ ol . /81 . 22 . 33 .44
+ o2 . /82 . 11 . 33 . 44
+ o3 . /83 . 11 . 22 . 44
+ 04 . /84 . 11 . 22 . 33
+ ol23 . /8123 . 44
+ ol24 . /8124 . 33
+ ol34 . /3134 . 22
+ o234 . 8234 . 11.
26—2
204
RECHEBCHES ULT^IRIEURES SUB LES D^TTERMINANTS QAUCHES.
[137
II est k peine n^essaire de remarqaer que dans les Pfaffians k droite, oil entrent
des symboles tels que la, fil, &c, qui ne se trouvent pas dans le determinant dont il
s'agit, il faut Airire la = — al, /91 = — 1/3, &c. Le symbole /8a ne se trouve ni dans le
determinant, ni au cdtd droit. Cependant il convient de poser /9a = — a/9 ; car cela extant,
il serait permis de transformer les P/affians, en ^rivant par exemple a/912 = — /3al2.
Je remarque en passant que, si avant de poser T^uation /9a s — a/9, on suppose que
les deux s}rmboles a, fi deviennent identiques (si par exemple on ^rit a = )3 = 5), on
aurait par exemple
a/8.12 = a/9.12 + al.2/9 + a2.i91 = 65.12 + 61.25 + 62.51 = 55.12, &c.,
et on retrouverait ainsi la formule pour le ddveloppement de 12346  12345.
La nouvelle formule pent Stre appliqu^e imm^diatement au d^veloppement des
determinants mineura. En effet, en se servant de la notation des matrices, on aura
11, 12, 13
21, 22, 23
31, 32, 33
— 1
123 123
+ 23 23,  13 23,  12 32
 23 I 13, + 13 I 13,  21 I 31
3'2Tl2, STTll, +12 I 12
11.
21.
31.
41,
12.
22,
32.
42,
13,
23,
33,
43,
14
24
34
44
— 1
1234 1234
+ 234 I 234,  134  234,  124  324,
 234 I 134, + 134  134,  214  314,
 324 I 124,  314  214, + 214  214,
423 I 123,  413  213, 412  312, +123fT23
123 1
423
213
413
312 1
412
et ainsi de suite. On suppose toujours que chaque terme de la matrice k droite soit
divis^ par le ddnominateur commun. On voit que les determinants mineurs qui cor
respondent k des termes tels que aa, sont des determinants gauches ordinaires, avec le
signe + , tandis que les determinants mineurs qui correspondent k des termes tels que
a/9, sont des determinants gauches hordds tels que /9...  a..., avec le signe — .
Pour donner des exemples de la verification de ces formules, je remarque que Ton
doit avoir
123 123= 11 . 23 23
 12 . 23 I 13
13 . 32 I 12:
equation qui pent aussi Stre ecrite sous la forme
123 123= 11 . 23 23
+ 21 . 23 13
+ 31 . 32 I 12
137] BECHEBCHES ULT^RIEURES SUB LES DJ^'ERMmANTS GAUCHES.
205
En effet, en d^veloppant les deux cdt^, on obtient:
11.22.33 + 11.(23)* + 22.(13)' + 33.(12)» = 11.(22.33 + (23)*)
+ 21.(21.33 + 23.13t)
+ 31 . (31 . 22 + 32 . 12t).
On Toit que les deux termes marqu^ par un f* se d^truisent et que I'^uation est
identiqtie. On doit avoir de mSme,
1234 I 1234 = 11 . 234
 12 . 234
 13 . 324
 14 . 423
ou, ce qui est la mSme chose :
1234 1234 » 11 . 234
+ 21 . 234
+ 31 . 324
+ 41 . 423
234
134
124
123,
234
134
124
123;
c'estadire, en d^veloppant des deux cdt^s:
11.22.33.44 + 11.22.(34)" + 11.33.(24)' + 11.44.(23)'
+ 22.33.(14)' + 22.44.(13)» + 33. 44. (12)' + (1234)' =
11 [22 . 33 . 44 + 22 (34)* + 33 (42)' + 44 (23)']
+ 21 [21 . 33 . 44 + 2134 . 34» + 23 . 13 . 44t + 24 . 14 . 33t]
+ 31 [31 . 22 . 44 + 3124 . 24* + 32 . 12 . 44t + 34 . 14 . 22t]
+ 41 [41 . 22 . 33 + 4123 . 23* + 42 . 12 . 33t + 43 . 13 . 22t].
Cette expression est en effet identique, comme on le voit en observant que les
^^^ termes marqu^ par un "f* se d^truisent deux k deux, et que les trois termes
ntxkjrqu^ par un (*) sont ensemble ^uivalents k (1234)*.
Je remarque que le nombre des termes du d^veloppement du determinant gauche
^^ toujours une puiasance de 2, et que de plus, ce nombre se r^uit k la moiti^,
6^ r^uisant k z6io un terme quelconque act. Mais outre cela, le determinant prend
^lu cette supposition la forme de determinant d'un ordre inferieur de I'unite. Je
<^iksidke par exemple le determinant gauche 123  123. En y faisant 33 = et en
Af^centoant, pour y mettre plus de clarte, tous les symboles, on trouve
123 I 123' = 11' . (23')' + 22' . (13')'.
206
RECHKRCHES ULT^RIEUBES SUB LES D^ERMHTANTB GAUCHBS.
[137
De la, en ^crivant
11 = 13'. 11', 12 = 11'. 23',
22 = 18' . 22',
on obtient
12 I 12 = 11 .22 + (12)'
= 11'.{22'.(13')' + ll'.(23')»).
12 I 12 = 11' . 123 I 123'.
c'esttidire
On a de mSme
1234 I 1234' = 11' . 22' . (34')» + 11' . 83' . (24')« + 22'33' (14')» + (1234')'
et de 1^, en ^crivant
11 = 14'. 11', 12 = 11.24', 23 = 1234',
22 = 14'. 22', 13 = 11'. 34',
33 = 14' . 33',
on obtient
123 I 123 = 11 . 22 . 83 + 11 . (28)' + 22 . (31)' + 33 (Uy
= 11' . 14' {22' . 83' . (Uy + (1234')' + 11' . 22' . (34')' + 11' . 33'
(24')'1,
c'estidire,
De mdme
123 I 123 = 11' . 14' . 1234 I 1234'.
12345 I 12345' = 11' . 22' . 33' . (45')'
+ 11' . 22' . 44' . (SSy
+ 11' . 33' . 44' . (25')'
+ 22' . 33' . 44' . (15')»
+ 11' . (2346')«
+ 22' . (1345')*
+ 33' . (1245')'
+ 44' . (lissy.
Or il est permis d'^rire
11 = 15'. 11', 12 = 11'. 25', 23 = 1235', 1234 = 2345'. 11'. 15',
22 = 15'. 22', 13 = 11'. 35', 24 = 1245',
33 = 15'. 33', 14 = 11'. 45', 34 = 1345',
44 = 15'. 44'.
En effet, les quantity k gauche ne sent li^ entre elles que par la seule ^na
tion 1234 = 12.34 + 13.42 + 14.23 qui est satisfiodte identiquement par les valeurs i^
substituer pour les quantity qui y entrent. Cela dtant, on trouve d'abord:
1234 I 1234 = 11' (16')' 12345  12345'.
137]
RBCHBBCHSS ULT^RIEUBES SUR LES DJ^TEBHINANTS OAUCHES.
207
Je prends encore on example. On a
123456 123456'= 11' . 22' . 33* . 44' .
(56')»
+ 11' . 22' . 33' . 55' .
(46')'
+ 11' . 22' . 44' . 55' .
(36')'
+ 11' . 33' . 44' . 55' .
(2&y
+ 22' . 33' . 44' . 55' .
(16')'
+ 11' . 22' . (U56y
+ 11' . 33' . (2456')'
+ 11' . 44' . (2356')»
+ 11' . 55' . (2346')'
+ 22' . 33' . (1456')«
+ 22' . 44' . (1356')'
+ 22' . 55' . (1346')"
+ 33' . 44' . (1256')'
+ 33' . 55' . (1246')*
+ 44' . 65' . (1236')'
+ (123456')».
mis d'^Tire:
11 = 16'. 11',
12 = 11'. 26', 23 = 1236', 34
= 1346'. 45
22 = 16' . 22',
13 = 11'. 36', 24 = 1246', 35
= 1356',
33 = 16'. 33',
14 = 11'. 46', 25 = 1256',
44 = 16'. 44',
15 = 11'. 66',
55 = 16'. 55',
1234 = 2346'. 11
'.16', 2345 = 128456'. 16',
1235 = 2366'. ir
.16',
1245 = 2456'. 11
' . 16'.
1345 = 3456'. 11
'.16;
= 1456',
car les valeun des quantity k gauche satisfont identiqaement aux ^uations qui ont lieu
entre ces mSmes quantity Par exemple I'^uation 1234 = 12.34 + 13.42 + 14.23
devient 2346'. 16'= 26'. 1346' + 63'. 1246' + 46'. 1236'.
Or I'expreasion k droite devient, en d^veloppant:
26' (13'. 46' + 14'. 68' + 16'. 34')
+ 63' (12' . 46' + 14' . 62' + 16' . 24')
+ 46' (12'. 36' + 13'. 62' + 16'. 28'),
208
RECHERCHES ULT^RIEUBES SUB LES D^ERMINANTS OAUCHES.
[137
et les termes qui contiennent le facteur 16', donnent ensemble 16'. 2346', les autres
terraes se d^truisent deux k deux. On obtient enfin, en effectuant la substitution:
12345 I 12345 » 11' . {IQ'y 123456  123456' ;
et ainsi de suite.
Je fisus les mdmes substitutions dans les matrices inverses, en supprimant cependant
la derni^re ligne et la demi^re colonne de chaque matrice. On trouve ainsi, en y
ajoutant les Equations cidessus trouv6es par rapport aux determinants:
13' . 123 I 123'
11' . 123 I 123' = 12 I 12,
+ 23 23',  13 I 23'
 23 I 13', + 13 I 13'
12 12
2 2 +
12 I 12
11
+2H,
. 12
+ lll
11' . 14' . 1234 I 1234' = 123  123,
14' . 1234 I 1234'
+ 234 I 234,  134  234,  124  324
2347~i24, + 134  134,  214  314
 324 I 124,  314  214, + 124  124
123 I 123
 23 I 23 +
123 I 123
11
 13 I 23,  12 I 32
+ 23 I 13,
+ 32T12,
+ 13 I 13,  21 I 31
 31 I 21, + 12 I 12
11' . (Wy . 12345 I 12345 = 1234  1234,
15' . 12345 I 12345'
+ 2345 I 2345',  1345  2345',  1245  3245',  1235  4235'
 2345 I 1345', +1346  1345',  2145  3145',  2135  4135'
 3245 I 1245',  3145  2145', +1245  1245',  3125  4125'
 4235 I 1235', 4135  2135',  4125  3125', + 1235  1235'
1234 1234
 234 I 234 +
1234 1234
11
+ 234 I 134,
+ 324T124,
+ 423 I 123,
et ainsi de suite.
 134 I 234,  124  324,  123  423
+ 134 I 134,  214 I 314,  213  413
 314 I 214, + 124 I 124,  312  423
 413 I 213,  412 I 312, + 123 I 123
137]
BECHERCHES ULTl^IEUBES SUR LBS D^TTEBMINAITTS OAUCHES.
209
II est bon de changer un peu la forme de ces ^uationa On en d^uit sans
peine
13' . 123 I 123'
2 . 23 I 23'  123 123*
11
,  2 . 13 I 23'
 2 . 23 I 13',
+ 2.13 I 13'
, 123 123'
22
12 12
2.2 2+
12 I 12
11
, 2.12
+ 2.2 1,
+ 2.1 1
12 12
22
^'.123411234'
2 . 234 234'
, 1234 1234'
11'
, 2.134 I 234',
2.124 324'
2 . 234 I 134',
2 . 134 134'
, 1234 1234'
22'
, 2 . 214 I 314'
2 . 324 I 124',
2.814 I 214',
..mnsi'.^*^
123 123
2 . 23 I 23
123 123
11
 2 . 13 23,
+ 2.23 I 13,
+ 2. 13 I 13
123 123
22
+ 2.32 ! 12,
 2 . 81 21,
2
,12
32
2
.21
31
•
123 1 123
+ 2.
12
12
33
et ainsi de suite.
Les formules que je viens de presenter, peuvent 6tre appliqu^ aussitdt k la
solution de la question suivante: "Trouver Xi, x,, x„ &c., fonctions lin^ires de Xi,
Xi, Xj, &c. telles que
11 Xi« + 22 x,»+ 33 x," + &c. = 11 a?i" + 22 a:,» + 33 a?,« + &c./'
c*estitdire : transformer une fonction quadratique llari' + 22ajj' + 33a:b' + &c. en elle
mSme par des substitutions lin^aires. II suffira d'^rire la solution pour le cas de
trois ind^termin^: on satisfait identiquement k I'^uation
11 Xi« + 22 x,» + 33 x,« = 11 a?i» + 22 a?,* + 33 a?,»
en ^rivant
1
(Xj, Xj, Xj) —
123 I 123
+2.23 I 23
123 123
11
, 2.13 I 23,
2.12 I 32
(ll«i, 22«,. 33a^).
2.23 I 13.
I2.327T2,
+ 2.13 I 13
123 123
22
, 2.21 131
2.31 I 21,
+2.12 12
123 I 123
33
C. II.
27
210
RECHEBCHE8 ULTI^EUIIES SUB LES Dl^TEBMINANTS GAUCHES.
[137
Voil^ la transformation propre. On en tire la transformation impropre de ll^i' + 22d!:,''
en dlemSme en ^rivant 33 = 0; car, cela pos^, les valeurs de Xi, x, ne contiennent
pas Xg, et Ton n'a plus besoin de la valeur de x,. On obtient ainsi la solution
suivante; savoir, on satis&it identiquement h T^uation
en ^rivant
(x,, Xj) =
123 I 123'
ir Xi» + 22^ x,» = ir a?i« + 22^ a?,«
2.23 23'^?^jL^', 2. 13 I 23'
(ira?», 22^:,),
 2 . 23 I 13',
2 . 13 I 13
, 123 I 123^
22
ce qui est une transformation impropre. Mais en y fiusant la substitution 11 =13'. 11',
22 = 13'. 22', 12 = 11'. 23', on r^uit la solution k celleci, savoir on satis&it identique
ment k r^uation llxi" + 22x,"= lla?i» + 2aci" en ^crivant
(xi, ^fd 12 I 12
1 O TO .
2.21 2 + i^^, 2.112
11
+ 2 . 21,
+2.11
12 I 12
22
(lla^i, 2ac),
ce qui est encore une transformation impropre, qui correspond de plus pr^ a la
formule pour la transformation propre; la seule diffi^rence est que les signes des
termes de la premiere colonne de la matrice en sent chang^
En introduisant des lettres simples a, 6, &c & la place des symboles 11, 22, &c.,
je consid&re d'abord la transformation propre
ax* + 6y* = da^ + 6y*.
Ici, en 6criyant
11.
12
^
a, p
21.
22
V, h
la formule donne
(X, y) =
ab + ^
abi^,  2i*
2va , ab'i^
(«. y)
La transformation impropre
ax* + 6y* = ewB* + 6y*
s'obtient au moyen de la formule donnee plus bas pour la transformation propre de
la fonction aa^ + bi/* + c^ en eUemSme, En y &srivant c = 0, on obtient
(X, y) =
aX« + 6^»
2\fia , — aX* + bfi*
(«i y)
137]
RBCHEBCHE8 ULT^RIEUIIES SUB LES D]fer£ItMINANTS GAUCHES.
211
J'ai d4jk £Edt voir de quelle mani&re cette formule se rattache k la formule pour
la transformation propre; la difif<^rence entre les formes de ces transformations dans ee
cas simple est assez firappante«
Pour obtenir la transformation propre
ax' + 6y' + cz* = cur* + 6y' + C2^,
j'&ris
11.
12,
13
=
21.
22.
23
31.
32.
33
v, 6,
X
c
eette formule donne
(x, y, z) =
abc + aK*^ bfA* + ci^
a6c + aX*6A*ci^, 2(\fjhcv)b , 2(vX + 6aa)c
2(XA+cv)a , abc — aX^ + bfjL^^cp', 2(jiv — a\)c
2 (vX — bfi) a , 2 (/LM/ + aX) 6 , a6c — aX* — 6/a' — ci^
La transformation impropre 0n eUemSme
ax' + 6y' + cz' = oa?" + 6y* + c^'
(^. y. z)'
^tre tir^ de la transformation propre en diemime de la fonction donn^e eiapr^s
f 6y' + C2^ + rft(;*; en y ^crivant d = 0, on obtient
^^* ^' ^^ 6cp«+ca<r» + a6T« + <^»
bcp^ ■{■ cao^ •\' abi*  4^,  2bT(l>  2bcp(r , 2ca<f>2bcTp
2ar4> — 2ac/!>cr , ic/^* — caa* + air" — 0*, — 2cp<l> — 2ca<rT
2aa(l> — 2abpT , 2bp4>2abaT , bcp* h caa* ^ abi* — (f)^
(«^» y, '^).
Pour verifier que cette expression n'est en effet autre chose que la formule pour
^ transformation propre, en y changeant les signes de tous les termes, j'^ris dans la
^^Hxiule pour la transformation propre, as=& = c = a>. On a ainsi pour la transformation
P^pre
x' + y* + z' = a;« + y* + 8^,
^ ^nation
(t.
cD«+X*/A*i'', 2XA2vai , 2i^X+2^cD
2X^ + 21^0 , ®»X* + /A«i;*, 2^1^2X01
2i/X2/*® , 2/Av + 2Xa> , ««X« + aa"»^
(^, y, '^),
27—2
212
RECHEBCHBS ULT^BIEnitES SUB LES Dl^EBMINANTS GAUOHES.
[137
et en ^rivant dans la formule pour la transformation impropre, a=& = c = l, d^^O, et
\, fi, V, — m au lieu de p, cr, r, <!>, on obtient pour la transfonnation impropre
r^uation
x« + y«+2? = a;« + y*+8*,
a)*X* + ^« + i^*, 2\A+2w , 2i^X2/*«
2X/i2i^a) , a)* + X*A** + i^, 2/iav2X»
2ifX + 2/u» , 2/iAv2\i» , ©« + X"At*i^
Pour obtenir la transformation propre
ax'  6y' + cz' + rfw' = flur* + fty' + ci* + du^,
11, 12, 13, 14
21, 22, 23, 24
31, 32, 33, 34
41, 42, 43, 44
oela donne d'abord, en mettant pour abr^ger,
la valeur du determinant
(^» y» ^)'
j'&ris
a,
Vf
M,
P
V,
6,
X,
<T
M,
X,
c,
T
P>
<r.
T,
d
— 1
1234 I 1234==aJcd + 6c/5« + ca<r* + a67* + acrx« + M/i» + c(ii;« + ^
(ce que je repr^sente par &).
J'ajoute aussi la valeur de la mairicA inverse
11, 12, 13, 14
21, 22, 23, 24
31, 32, 33, 34
41, 42, 43, 44
ocd + 6t* + c/E)* +dfi*,
ddK^rp^ + dfUf^aoTt
aca + fjLtf} ^ cvp +aXT,
bdfi H oif} + dXjf — bpT ,
— odK ^pif) + dfjtv — aoT,
abd + aa* + bf^ +dv^ ,
dbr + 1^^ + bfip — aXo",
savoir:
1
bed +bT* +ca* +(iX«,
k
cdp + T^ + dXft — cpa.
— Mft — cr^ + (iXi^ — bpr,
bcp +X^ + ci'cr ^bfir,
— 6cp — X^ + cva — 6/iT
• aca — fuf) — cpp 4aXT
— air — 1^0 + 6/Lif) — a\a
abc +aX'+6/i« + ci/»
137]
BBCHERCHES ULT^RIEURES SUR LES Dl^RMINANTS GAUCHES.
213
On a pour la transformation, I'^uation (x, y, z, w) =
1
k
abed — 6cp' + ccur* + abi^h adX?
2a (cdv + T0 + dXfjk — cp<r) ,
2a (— bdfi — cr^ + dKv — bpr),
2a (bcp + X^ + cvtr — bfir) ,
26 (— cdv — T^ + dKfi — cpcr) ,
a6c(i + bcp* — ccur* + abr* — ad\"
26 (adk + p^ + rf^v — a<rr) ,
26 (oca + fjul> — cvp + aXr) ,
2c (6d/i + cr^ + (iXv — 6pT) ,
2c (— adX — p<l> + dfjkv — cutt) ,
a6cd + bcp^ + ccur" — a6T' — odX*
 6d^« + cdi/*  0«
2c (a6T + 1^ + bfjLp — aXo*) ,
2d (— 6cp — X0 4 ci'cr — bfir)
2d (— accr — i^ — ci'p + aXr)
2d (— abr — i^^ + 6/ip — aXo")
abed — 6c/!)' — caa* — abr* 4 adX*
(or, y, J?, m;).
Je suppose que Ton ait a = 6 = c = d = «, et j'&jris ^ = , c'estidire
^ = — ^ ^*^ — ^ ou \p + /xo + in + '^a) = 0. En fiedsant cette substitution, on trouve
d'abord k — fu^B, oh
iJ = X« + ;i« + l;« + ^ + p« + <T« + T»+a)«,
et puis pour la transformation propre
x« 4 y « 4 z« + w' = a* + y ' + 8^ + W,
TAjuation (x, y, z, w) =
p« + <r* + T« + tt)* + X"^"I^'^,
2a>v  2t'^ + 2X/i  2/!)cr,
 2®/i + 2<r^ + 2Xi/ 2/yr,
2a>p  2X^ + 2i^cr  2/iat,
1
 2o>v + 2t^ + 2X/i  2pcr,
2a)X  2p'^ + 2^1^  2<rr,
2a>(r  2/i^  2i;/o + 2Xt,
2mp.  2<r^ + 2Xv —2pT
2oiX + 2p^ + 2/u/ 2<rr
p« + <r»  T» + «» X« AA« + i^
2toT 2i^+ 2aap  2Xcr
^, 
2aip + 2X^ + 2va  2/iT
2a)cr + 2^^  2pp + 2Xt
2ftrr + 2ir^ + 2/i/)  2X<t
^'
(a:, y, ^, w).
On peut changer la forme de cette expression, en y &rivant
X = i(«aO. /t = i(/9i8'). i' = i(77'). V^ = i(S8').
214
BECHEBCHES ULTl^RIEUBES SUR LES Dl^TEBMINANTS GAUCHES.
[137
cela donne
de mani^re qu'en ^crivant
on obtient
R = \{M\M')^.J{MM'),
et la formule pour la transformation devient
(x, y, z, w) =
V {MM')
aa' + i8/8' + 77' +S8',
a/8'i8a'+78'+V.
cu^fifi'+yy'+iS',
aS'  ^'  yfi'  Sc^ .
»y'y8S' + 7««'Si8',
»y' +/SS'
oS' —fiy'
yd  1?,
y^ + Scl.
77' + SS*,
7S'  S7'.
oy + ies* + 70' + s^
afi'ficl+y^ + Sy'
ad ^ffyy'+BS'
(x. y, z, w).
la
Oil voit done que mdme sans supposer T^uation M = M\ cette formule donne
transformation propre
x« 4 y« 4. z* + w* = a;* + y* + ^ + ?*;".
Cette solution est k peu pr^ de la m6me forme que la solution impropre donn^
par Euler dans son m^moire "Problema algebraicum ob affectiones prorsus singulares
memorabile " Nov. Comm. Petrop., t. xv. 1770, p. 75, et Coram. Arith. collects, [4to.
Petrop. 1849], t. i. p. 427. Je remarque aussi que cette ' m6me solution pent 6tre
d^uite de la th^orie des Quaternions, En effet, t, j, k ^tant des quantity imaginaires
telles que i*=j' = A;* = — 1, jfc = — ^' = t, ki — — ik^jy %j^''ji=sk, on obtient, en effectuant
la multiplication:
X, y, z, w ayant les mdmes valeurs que dans la demi^re formule de transformation.
En changeant les signes des termes de la quatri&me colonne, on en tire pour la
transformation impropre
x' + y' + z' + w* = a:' + y* + »• + w*.
137]
RECHERCHES ULTiRIEURES SUR LES D^^TERMINANTS OAUCHES.
215
la fonnule suivante plus sym^trique :
(X, y, z, w)^^^j^^
oa' +/9/8'+ry' + S8', ciff
ev9'/8a'+78'&y', aa'
»y'y88'7a'+S/8', US'
/9a'  78' + 87' ,
aiSf fir/
ettt'+ff^
afffia'
To'S/y,
77'+SS'.
7«'  V,
ay  ^87' + 7)8'  So'
a7' ffS' yc^ SIS'
afi'+fic^ yB' By'
a«'+y8/8' + 77'88'
(a;, y, «, w).
Ces formules pour la transformation, tant propre qu'impropre, de la fonction
^ + y* + '8* + W en eUemSme, sont utiles dans la th^rie des polygenes inscrits dans
line sur£Eu» du second ordre.
216
[138
138.
RECHERCHES SUR LES MATRICES DONT LES TERMES SONT
DES FONCTIONS LINEAIRES D'UNE SEULE INDfiTERMINEE.
[From the Jounial fiir die reine und angewandte Mathematik (Crelle), torn. L.
(1855). pp. 313—317.]
Je pose la matrice
A y IJ I G , ...
A I IJ , G , ...
dont les termes {n* en nombre) sont des fonctions lin^aires d'une quantity «, et je
considfere le determinant form^ avec cette matrice, et les determinants mineurs fonn&
en supprimant un nombre quelconque des lignes et un nombre ^gal de colonnes de
la matrice. En supprimant une sevle ligne et une aeule colonne, on obtient les
premiers mineurs; en supprimant deux lignes et deux colonnes, on obtient les seconds
mineurs; et ainsi de suite. Cela ^tant, je suppose que la quantity s a ^t^ trouvde
en egalant k z^ro le determinant form^ avec la matrice donn^e ; ce determinant sera
une fonction de s du ni^me degre qui g^neralement ne contiendra pas de facteurs
multiples. On voit done qu'un facteur simple du determinant ne pent pas entrer comme
facteur dans les premiers mineurs (c'estkdire dans tous les premiers mineurs) ; mais
en supposant que le determinant ait des facteurs multiples, un fisu^teur multiple du
determinant pent entrer comme facteur (simple ou multiple) dans les premiers mineurs,
ou dans les mineurs d*un ordre plus eieve. II importe de trouver le degre selon lequel
un facteur multiple du determinant pent entrer comme facteur des premiers mineurs,
ou des mineurs d'un ordre quelconque donne.
Cela se fait trfes facilement au moyen d'une propriete generale des determinants;
si les mineurs du (r4l)ifeme ordre contiennent le facteur («— a)* (c'estitdire, si tous
138] BECHEBCHES SUB LES MATBICES &C. 217
les mineurs de cet ordre contiennent le facteur (« — a)*, mais non pas tous les fistcteurs
(» — a)*"*"*) ; et si de m^me les mineurs du rifeme ordre contiennent le fiskcteur (« — a)^ ;
alors les mineurs du (r — l)i&me ordre contiendront au moina le £Etcteur (« — a)^^"*.
^utrement dit: les mineurs du (r — l)ifeme ordre contiendront le facteur (« — a)^ od
'y > 2)8 — a, ou, ce qui est la mfime chose, a — 2/8 + 7 <t ; c'estkdire : en formant
la suite des indices des puissances selon lesquelles le facteur {s — a) entre dans les
Tuineurs premiers, seconds, &c. (il va sans dire que cette suite sera une suite dScroissante),
lee di£f(^rences secondes seront positives [c'estkdire non nigativ€s\ Je repr^sente par
^<> /8, 7, ... la suite dont il s'agit; je suppose, pour fixer les iddes, que h soit le
ciemier terme qui ne s'^vanouisse pas, et j'^cris
a, ^, 7, S, 0, 0, . . .
aA )87, 7S, S, 0,...
a2;9 + 7. i827 + S, 72S, S, 0,...;
xci, quel que soit le nombre des termes, tous les nombres de la troisi^me ligne seront
positi&, et en reprdsentant ces nombres par /, /', /'', &c., on obtient:
a=/+2/' + 3r + 4r+...,
^= /'+2r+3r+...,
7= /+2r+...,
n y a ici k consid^rer que le nombre a, indice de la puissance selon laquelle le
facteur {s — a) entre dans le determinant, est donn^ ; il sera done permis de prendre
jy>ur /, /', /",... des valeurs enti^res et positives quelconques (z^ro y compris) qui
aatisfont k la premiere ^nation; les autres ^nations donnent alors les valeurs de
iS, % Sy &c On forme de cette mani&re une table des particularity que pent printer
un facteur multiple {s — a)* du determinant ; cette table est compos^e des symboles
^> /3, y,..., et les nombres a,l3,.., de chaque symbole font voir le degr^ selon lequel
le £acteur (s — a) entre dans les determinants, dans les mineurs premiers, seconds, &c.
Or il est trfes facile de former, au moyen des tables pour a^l, a = 2, ... as A;, la table
pour a=sit+l. On a par exemple pour a=l, a = 2, a = 3, a = 4t les tables suivantes:
Pour a = 1, 1.
Pour = 2, 2, 21.
Pour a = 3, 3, 31, 321.
Pour = 4, 4, 41, 42, 421, 4321.
De \k on tire la table pour a = 5, savoir :
Pour a = 6, 6, 61, 62, 621, 631, 6321, 64321.
£n eifet, le premier terme est 6, et on obtient les autres termes en mettant le nombre
) devant les symboles des tables pour a^l, 0=2, a = 3, a = 4, en ayant seulement soin
C. II. 28
218 RECHGRCHGS SUR LES MATRICES DONT LES TBBHES 30NT [l38
de supprimo les symbolee 58, 54, 541, 542, 5421 pour lesquels le premier terme de
la Buite des diffdreDces eecondes est nigsAif. On trouve de mSme pour a = 6, la table
suivante, savoir :
Pour a = 6
6, 61, 62, 621, 63, 631, 6321, 642, 6421, 64321, 654321;
et ainsi de suite. Lea nombrea dea Bymboles pour a = l, 2, 3, 4, 5, 6, 7, 8, &c sont
1, 2, 3, 5, 7, 11, 15, 22, [30, 42, 56], &c.; ce aont les coefficients des pmBSonces a^. of, a?
&C. dans le d^veloppement de
(1ic)' (la:')' {lx'y* (13)' (l«^)' ... &c.
foQCtions qui se pr^entent, comme on sait, daos la th^rie de la pari^on des nombrea.
Maintenant, au lieu de coosiddrer un seulfacteur du determinant, J e considfere toils
les facteurs: par exemple pour n = 4, le determinant pout avoir un fiacteur double (s—df,
et un autre facteur double (« — 6)* ; il peut de plus arriver que le facteur (« — a) soit fiwteur
simple des premiers mineurs, mais que le facteur (s ~ h) n'entre paa dans lee premiers
mineurs. Le eymbole qui correspond au &cteur (s — a) sera 21, et le symbole qui
coirespond au facteur (B—h) sera 2. En combinaot cee deux symboles, on aura le
symbole composd
qui denote que le determinant a deux facteurs doubles de
la clasee dont il s'agtt. Je forme de ces symboles composes
n = 2, n = 3, 7i = 4, &c. On a:
table* pour n = 1 ,
^0'
1
2
. 21
1
1
1
_
. 31 , 321
Pour
n =
4 :
1
2
21
3
321
2
21
1
1
1
1
1
1
1
1
2
2
^
et ainsi de suite.
g.g,
42 . 421 , 4321 [
138] DES FONCTIONS LINJ&AIRES d'UNE SEULE INDlfeTERMINilE. 219
321
Pour donner encore un exemple du sens de ces symboles, le symbole denote
cjue le determinant a un facteur (s — a) qui entre comme facteur triple dans le deter
minant, comme &eteur double dans les premiers mineurs, et comme facteur simple dans
les seconds mineurs; Tautre facteur du determinant est un facteur simple (s — b), Les
xiombres des symboles pour n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, &c. sont 1, 3, 6,
14, 27, 58, 111, 223, 424, 817, 1527, &c. ; ces nombres sont les coefficients de oc", a?, a*,
<&c, dans le developpement de
( 1  jr)^ (1  aj»)« (1 a^)'(l ic*)» (1  a^y (1  a:*)"" (1  a^)"(l  af)^ (1  a;»)». . . &c.
cju les indices 1, 2, 3, 5, 7, 11, &c. forment la suite qui se pr^sente dans la theorie
c^e la partition des nombres, dont j'ai parie plus haut II est tr^s facile de d^montrer
f^u'il en est ainsi.
Les r^sultats que je viens de presenter sont en partie dus k M. Sylvester (voyez
isson m^moire "An enumeration of the contacts of lines and surfaces of the second
cz^rder," Philosophical Magazine, [voL I. (1851), pp. 18 — 20]). En efFet, M. Sylvester
ciommence par etendre h. des fonctions d*un nombre quelconque d'indeterminees Tid^e
^^ometrique des contacts des courbes et des surfaces. £n considerant les deux ^qua
't.ions quadratiques [7=0, F=0, il forme le discriminant de la fonction quadratique
ZJ \ aV^ et ir cherche dans quel degr^ chaque facteur de ce discriminant pent entrer
c^romme facteur dans les mineurs premiers, seconds, &c. Le discriminant de M. Sylvester
<=^st un determinant sym^trique ; mais cela ne change rien k la question, et je n'ai
fjEiit que reproduire Tanalyse de M. Sylvester, en donnant cependant Talgorithme pour
la formation des symboles, et de plus la loi pour le nombre des symboles. M. Sylvester
cJonne pour n = 2, 3, 4, 6, 6, des nombres qui, en ajoutant k chacun le nombre 2,
pK>ur embrasser deux cas extremes qui ne sont pas comptes, seraient 3, 6, 14, 26, 58.
X\ se trouve dans le nombre 26 une erreur de calcul ; ce nombre devrait 6tre 27, et
^n suppliant le premier terme 1, on a la suite trouv^e plus haut, savoir 1, 3, 6, 14;
27, 58, &c. ; il y a de mSme une erreur de calcul dans les nombres donnas par M.
Sylvester pour n = 7 et n = 8.
Mais tout cela s'applique k une autre theorie g^ometrique, savoir k la theorie des
figures homographes. Pour fixer les id^es, je ne considere que les figures dans le plan.
Kn supposant que x, y, z soient les coordonnees d*un point, et en prenant pour (x, y, z)
cles fonctions lin^aires de (ar, y, z) on aura (x, y, z) comme coordonnees d'un point
honiographe au point (x, y, z). En cherchant les points qui sont homographes chacun
^ soimfime, on est conduit aux Equations x— «a? = 0, y — «y = 0, z — «z = 0. Les
c^uantit^s k gauche x — sx, y — «y, z— sz sont des fonctions lindaires de x, y, z, ayant
pour coefficients des fonctions lin^aires de s. On a ainsi une matrice dont les termes
^ont des fonctions lin^aires de s] la th^orie enti^re se rattache aux propriet^s de
eette matrice. Pour le cas gdndral de Vhomographie ordinaire, on a le symbole
28—2
220
RECHERCHES SUB LES MATRICES &C.
[138
1
1
1
, pour Yhomologie, le symbole
; les autres symboles
2
1
• IB
31
se
321
au cas de Tidentit^
rapportent k des cas moins g^draux, et le symbole
complete des deux figures; y compris ce caslimite de Tidentit^ complete, il eziste
pour le plan 6 espfeces d'homographie ; pour Yespace ordinaire il existe 14 especes
dliomographie. Je reviendrai k cette th^rie k une autre occasion.
LtmdreSy le 24 Mai 1854.
139]
221
139.
AN INTRODUCTOKY MEMOIR UPON QUANTICS.
[From the Philosophical Transactions of the Royal Society of London, vol. cxliv. for the
year 1864, pp. 244—258. Received April 20,— Read May 4, 1854.]
1. The term Quantics is used to denote the entire subject of rational and integral
fimctiens, and of the equations and loci to which these give rise ; the word " quantic "
is an adjective, meaning of such a degree, but may be used substantively, the noun
understood being (unless the contrary appear by the context) function; so used the
word admits of the plural •' quantica"
The quantities or symbols to which the expression "degree " refers, or (what is the
same thing) in regard to which a function is considered as a quantic, will be spoken
of as "facients." A quantic may always be considered as being, in regard to its
facients, homogeneous, since to render it so, it is only necessary to introduce as a
facient unity, or some symbol which is to be ultimately replaced by unity; and in the
cases in which the facients are considered as forming two or more distinct sets, the
quantic may, in like manner, be considered as homogeneous in regard to each set
separately.
2. The expression "an equation," used without explanation, is to be understood as
meaning the equation obtained by putting any quantic equal to zero. I make no
absolute distinction between the words "degree" and "order" as applied to an equation
or system of equations, but I shall in general speak of the order rather than the
degree. The equations of a system may be independent, or there may exist relations
of connexion between the different equations of the system; the subject of a system
of equations so connected together is one of extreme complexity and difficulty. It will
be sufficient to notice here, that in any system whatever of equations, assuming only
that the equations are not more than sufficient to determine the ratios of the facients,
and joining to the system so many linear equations between the facients as will render
the ratios of the facients determinate, the order of the system is the same thing as
the order of the equation which determines any one of these ratios; it is clear that
for a single equation the order so determined is nothing else than the order of the
equation.
222 AN INTRODUCTORY MEMOIR UPON QUANTICS. [l39
3. An equation or system of equations represents, or is represented by a locus.
This assumes that the facients depend upon quantities x^ yi ••• the coordinates of a
point in space ; the entire series of points, the coordinates of which satisfy the equation
or system of equations, constitutes the locus. To avoid complexity, it is proper to take
the facients themselves as coordinates, or at all events to consider these facients as
linear functions of the coordinates; this being the case, the order of the locus will be
the order of the equation, or system of equations.
4. I have spoken of the coordinates of a point in space, I consider that there is
an ideal space of any number of dimensions, but of course, in the ordinary acceptation
of the word, space is of three dimensions; however, the plane (the space of ordinary
plane geometry) is a space of two dimensions, and we may consider the line as a space
of one dimension. I do not, it should be observed, say that the only idea which can
be formed of a space of two dimensions is the plane, or the only idea which can be
formed of space of one dimension is the line ; this is not the case. To avoid complexity,
I will take the case of plane geometry rather than geometry of three dimensions; it
will be unnecessary to speak of space, or of the number of its dimensions, or of the
plane, since we are only concerned with space of two dimensions, viz. the plane ; I say,
therefore, simply that x, y, z are the coordinates of a point (strictly speaking, it is the
ratios of these quantities which are the coordinates, and the quantities x, y, z themselves
are indeterminates, i.e. they are only determinate to a common factor prhs, so that in
assuming that the coordinates of a point are a, )8, 7, we mean only that x : y : z^a : 0:y,
and we never as a result obtain x, y, z = a, /3, 7, but only x : y : z^a : /3 : y; but
this being once understood, there is no objection to speaking of x, jf, z as coordinates).
Now the notions of coordinates and of a point are merely relative; we may, if we
please, consider x : y : z as the parameters of a curve containii^ two variable para
meters; such curve becomes of course determinate when we assume x : y : z^a : fi : y,
and this very curve is nothing else than the point whose coordinates are a, 0, 7, or
as we may for shortness call it, the point (a, /3, 7). And if the coordinates (x, y, z) are
connected by an equation, then giving to these coordinates the entire system of values
which satisfy the equation, the locus of the points corresponding to these values is the
locus representing or represented by the equation ; this of course fixes the notion of a
curve of any order, and in particular the notion of a line as the curve of the first
order.
The theory includes, as a very particular case, the ordinary theory of reciprocity in
plane geometry; we have only to say that the word "point" shall mean "line,* and the
word "line" shall mean "point," and that expressions properly or primarily applicable
to a point and a line respectively shall be construed to apply to a line and a point
respectively, and any theorem (assumed of course to be a purely descriptive one) relating
to points and lines will become a corresponding theorem relating to lines and points;
and similarly with regard to curves of a higher order, when the ideas of reciprodtv
applicable to these curves are properly developed.
5. A quantic of the degrees m, m'... in the sets (a?, y...), {x\ y'...) &a will for the
most part be represented by a notation such as
m m'
(♦$^, y...$a:', y'... )...).
139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 223
^where the mark * may be considered as indicative of the absolute generality of the
qaantic; any such quantic may of course be considered as the sum of a series of
therms afy^...fl/*y^..., &c. of the proper degrees in the different sets respectively, each
term multiplied by a coefHcient ; these coefficients may be mere numerical multiples
of single letters or elements such as a, b, c,..., or else functions (in general rational
siod integral functions) of such elements ; this explains the meaning of the expression
*' the elements of a quantic": in the case where the coefficients are mere numerical
xnnltiples of the elements, we may in general speak indifferently of the elements, or
of the coefficient& I have said that the coefficients may be numerical multiples of
single letters or elements such as a, b, c, ...; by the appropriate numerical coefficient
of a term afy^...a:'*y^ ..., I mean the coefficient of this term in the expansion of
m
(a: + y...) (a/ + y'... )...);
^nd I represent by the notation
m m
(a, b,...^x, y,...$J?', /,...)...)>
quantic in which each term is multiplied as well by its appropriate numerical coeffi
cient as by the literal coefficient or element which belongs to it in the set (a, 6,...) of
literal coefficients or elements. On the other hand, I represent by the notation
m m
(tt, b,...\x, y,...^a?', /,...)...)»
quantic in which each term is multiplied only by the literal coefficient or element
"vvhich belongs to it in the set (a, &,...) of literal coefficients or elements. And a like
distinction applies to the case where the coefficients are functions of the elements
(a, 6, ...).
6. I consider now the quantic
« m'
^Knd selecting any two facients of the same set, e.g. the facients x, y, I remark that
t^here is always an operation upon the elements, tantamount as regards the quantic
t:o the operation ady; viz. if we differentiate with respect to each element, multiply
V)y proper functions of. the elements and add, we obtain the same result as by differ
entiating with 3y and multiplying by x. The simplest example will show this as
"^rell as a formal proof; for instance, as regards Saa^ ^ bxy \ 5cy^ (the numerical
Coefficients are taken haphazard), we have ^a + lOcd^ tantamount to xdy', as regards
c^ (a: — ay) (^ — i8y ), we have — a(a+/8)3a + a^«4j8^^ tantamount to ady, and so in any
cither case. I represent by [ady] the operation upon the elements tantamount to aSy,
s^nd I write down the series of operations
'Where x, y are considered as being successively replaced by every permutation of two
clifferent £Eu;ients of the set (a:, y,...); ^> }/ as successively replaced by every permutation
of two different facients of the set {x\ y',...)> fti^d so on; this I call an entire system, and
224 AN INTEODCJCTORY MEMOIB UPON QUANTIC8. [l39
I say that it is made up of partial systems ooiresponding to the different fiunent sets
respectively; it is clear from the definition that the quantic is redaoed to zero by
each of the operations of the entire system. Now, besides the quantic itself, there
are a variety of other functions which are reduced to zero by each of the operations
of the entire system; any such function is said to be a oovariant of the quantic, and
in the particular case in which it contains only the elements, an invariant. (It would
be allowable to define as a covariant quoad any set or sets^ a function which is reduced
to zero by each of the operations of the corresponding partial system or systems, but
this is a point upon which it is not at present necessary to dwell)
7. The definition of a covariant may however be generalized in two directions:
we may instead of a single quantic consider two or more quantics; the operations
{aSy], although represented by means of the same symbols x, y have, as regards the
different quantics, different meanings, and we may form the sum 2 {^y}> where the
summation refers to the different quantics: we have only to oondder in place of the
system before spoken of, the system
2{a3y} — aj9y, ... ; 2{a:'3y'} —x'd^y ... &c. &c.,
and we obtain the definition of a covariant of two or more quantics.
Again, we may consider in connexion with each set of £Eu;ients any number of
new sets, the facients in any one of these new sets corresponding each to each with
those of the original set; and we may admit these new sets into the covariant. This
gives rise to a sum 8[a!d^y where the summation refers to the entire series of cor
responding sets. We have in place of the system spoken of in the original definition,
to consider the system
[ocdy] 'S{aSy\ ... [x'd^] — 8 {afd^\ ... &c. Ac.,
or if we are dealing with two or more quantics, then the system
2 [xdy] — S{ady), ... ; 2 {^'Sy'} 'S(x'd^), ... &c. &c.,
and we obtain the generalized definition of a covariant.
8. A covariant has been defined simply as a function reduced to zero by each oi
the operations of the entire system. But in dealing with given quantics, we may
without loss of generality consider the covariant as a function of the like form with
the quantic, Le. as being a rational and integral function homogeneous in regard to
the different sets separately, and as being also a rational and integral function of the
elements. In particular in the case where the coefficients are mere numerical multi
ples of the elements, the covariant is to be considered as a rational and integral
function homogeneous in regard to the different sets separately, and also homogeneous
in regard to the coefficients or elementa And the term "covariant" includes, as already
remarked, "invariant."
It is proper to remark, that if the same quantic be represented by means of different
sets of elements, then the symbols {xdy} which correspond to these different forms
139J AN INTRODUCTORY MEMOIR UPON QUANTICS. 225
of the same quantic. are mere transformatioDs of each other, i.e. they become in virtue
of the relations between the different sets of elements identical.
9. What precedes is a return to and generalization of the method employed in the
iirst part of the memoir published in the Camh. Math, Jour,, t, iv. [1845], and Camb.
a,nd DiM. Math. Jour., t, I. [1846], under the title "On Linear Transformations," [13
WLnd 14], and Crelle, t. xxx. [1846], under the title "M^moire sur les Hyperd^termi
iiants," [*16], and which I shall refer to as my original memoir. I there consider in
fact the invariants of a quantic
linear in regard to n sets each of them of m facients, and I represent the coefEcients
of a term Wrj/tZt... by rst...; there is no diflSculty in seeing that a, 13 being any two
<lifferent numbers out of the series 1, 2, ...m, the operation {^/sda;J is identical with the
operation
^S... last,., jr^,
^%vhere the summations refer to «, ty... which pass respectively from 1 to m, both inclu
^ve; and the condition that a function, assumed to be an invariant, i.e. to contain
*jnly the coeflScients, may be reduced to zero by the operation {«/i9ajJ — a?^aj«» is of
«30urse simply the condition that such function may be reduced to zero by the opera
tion {^/i^xj ; ^he condition in question is therefore the same thing as the equation
cDf my original memoir.
10. But the definition in the present memoir includes also the method made use
CDf in the second part of my original memoir. This method is substantially as follows:
^isonsider for simplicity a quantic £/' =
^i^ntaining only the single set {x, y...), and let CT,, ^, ... be what the quantic becomes
"Vrhen the set {x^y ...) is successively replaced by the sets (a:,, yi, ...), (ar„ yj,...), ... the
Xiumber of these new sets being equal to or greater than the number of facients in
t;he set. Suppose that A, B, C7, ... are any of the determinants
^«ii ^af> ^«,> •••
9yi> 9y»> 9y,>
^hen forming the derivative
'^here p, q, r ... are any positive integers, the function so obtained is a covariant in
xrolving the sets (a?i, yi,...)> (^f> yf>«««) ^m ^^^ if *ft®r *^h® diflferentiations we replace
c. n. 29
226 AN INTRODUCTORY MEMOIR UPON QU ANTICS. [l39
these sets by the original set {x, y, ...)* ^^ hAwe a co variant involving only the original
set (x, y, ...) and of course the coefficients of the quantic. It is in £act easy to show
that any such derivative is a covariant according to the definition given in this
Memoir. But to do this some preliminary explanations are necessary.
11. I consider any two operations P, Q, involving each or either of them differeD
tiations in respect of variables contained in the other of them. It is required to
investigate the effect of the operation P . Q, where the operation Q is to be in the
first place performed upon some operand tl, and the operation P is then to be per
formed on the operand Qfl. Suppose that P involves the differentiations da, ^,... in
respect of variables a, b, ... contained in Q and il, we must as usual in the operatioD
P replace da, dj,,.. by da + d'a, 96 + d^6> ••• where the unaccentuated symbols operate
only upon il, and the accentuated symbols operate only upon Q. Suppose that P is
expanded in ascending powers of the symbols 3^., 9'6, ...f viz. in the form PfPj + Pi + Ac.,
we have first to find the values of PjQ, PtQ, &c., by actually performing upon Q as
operand the differentiations 9'a, 9'6« The symbols PQ, P,Q, P,Q, &c. will then contain
only the differentiations d^, d^, ... which operate upon il, and the meaning of the ex
pression being once understood, we may write
P.Q = PQ + P,Q + P,Q + &c.
In particular if P be a linear function of 9a, dt, ..., we have to replace P by P + Pi,
where P, is the same function of d'a, S't. ••• that P is of 3., 96, ..., and it is therefore
clear that we have in this case
P.Q = PQ + P(Q).
where on the righthand side in the term PQ the differentiations 9., 96,... are con
sidered as not in an}rwise affecting the symbol Q, while in the term P(Q) these
differentiations, or what is the same thing, the operation P, is considered to be per
formed upon Q as operand.
Again, if Q be a linear function of a, 6, c, ..., then PiQ = 0, PiQ = 0, Ac, and
therefore P.Q = PQ^PiQ\ and I shall in this case also (and consequently whenever
P,Q = 0, P,Q = 0, &c.) write
P.Q^PQ^PiQl
where on the righthand side in the term PQ the differentiations 9a, 96,... are con
sidered as not in anywise affecting the symbol Q, while the term P(Q) is in each case
what has been in the first instance represented by PiQ.
We have in like manner, if Q be a linear function of 9a, 96, 9c, ..., or if P be
a linear function of a, 6, c, . . . ,
Q.P = QP + QiP);
and from the two equations (since obviously PQ^QP) we derive
P.QQ.P = P(Q)Q(P),
which is the form in which the equations are most frequently useful.
139] AN INTRODUCTORY MEMOIR UPON QU ANTICS. 227
12. I return to the expression
and I suppose that after the dififerentiations the sets (or,, yi, ...), (^9, ys> •••)> ^^' ^^^
replaced by the original set (x, y, ...). To show that the result is a co variant, we must
prove that it is reduced to zero by an operation 39 =
It is easy to see that the change of the sets (x^, yi, ...), (x^ yji •••)» ^^ i^^ ^^^ original
set (Xf y, ...) may be deferred until after the operation iD, provided that aSy is replaced
by ^9y, + ^y,4..., or if we please by Sxdy; we must therefore write ^ = {xidy} — Sxdy.
Now in the equation
il.BB.il=il(B)B(il),
where, as before, A (W) denotes the result of the operation A performed upon iB as
operand, and similarly ^(A) the result of the operation 39 performed upon A as
operand, we see first that A (W) is a determinant two of the lines of which are
identical, it is therefore equal to zero; and next, since 39 does not involve any
differentiations affecting A, that 39 (A) is also equal to zero. Hence il . 39 ~ 39 . il =
or A and 39 are convertible. But in like manner 39 is convertible with B, 0, &c.,
and consequently B is convertible with A^B^C^.... Now 391/itr,... =0; hence
m.APB^C'...L\U,...^0,
or ApB^O"... I/it/,... is a covariant, the proposition which was to be proved.
13. I pass to a theorem which leads to another method of finding the covariants
of a quantic. For this purpose I consider the quantic
m m
(♦$«?, y...\af, y'.. .)...),
the coeflScients of which are mere numerical multiples of the elements (a, 6, c, ...); and
in connexion with this quantic I consider the linear functions ^x\r)y..., fa/ + i;y"«>
which treating (f, 17,...), (f, V»'«)> *c ®* coeflScients, may be represented in the form
(f, 17, ...$ar, y, ...\ (f , 17', ...$a?', y', ...),...
we may from the quantic (which for convenience I call U) form an operative quantic
(♦$f, 17,... $f, V,...))
(I call this quantic 6)» the coeflScients of which are mere numerical multiples of
da, dt, 9«, •••> and which is such that
ie. a product of powers of the linear functions. And it is to be remarked that as
regards the quantic 6 and its covariants or other derivatives, the symbols da, df,, da...
are to be considered as elements with respect to which we may differentiate, &c.
29—2
228 AN INTRODUCTORY MEMOIR UPON QUANTICS. [l39
The quantic B gives rise to the symbols {^^], &c. analogous to the symbols {^y, &c.
formed from the quantic U. Suppose now that 4> is any quantic containing as well
the coefficients as all or any of the sets of 6. Then [xdy] being a linear function of
a, b, Ci... the variables to which the differentiations in ^ relate, we have
again, [rfi^] being a linear function of the differentiations with respect to the variables^
da, dtt dc,... in 4>, we have
these equations serve to show the meaning of the notations ^({^y}) and {n9(} (4>X an
there exists between these symbols the singular equation
14. The general demonstration of this equation presents no real difficulty, but toKn
avoid the necessity of fixing upon a notation to distinguish the coefficients of the
different terms and for the sake of simplicity, I shall merely exhibit by an exampl
the principle of such general demonstration. Consider the quantic
Cr= flur» + 36a;"^ + 3cy» + dy»,
this gives = pda + ^hfii, + ^rfde + rfd^ ;
or if, for greater clearness, da, dt, 9c, da are represented by a, fiy 7, S, then
and we have {aSy] = 369a + 2c96 4 dde,
and {nd(} = 3a9^ + 2^89^ 4 79a.
Now considering 4> as a function of 9a, 9^, 9e, da, or, what is the same thing, ol
a, A 7, S, we may write
4> {{xdy}) = a> (36a + 2c/8 + c^) ;
and if in the expression of <P we write a + 9a, ff^d^, 7 + 9c, 8 + 9^ for a, /8, 7, S (whe
only the symbols 9a, d^, de, da are to be considered as affecting a, 6, c, ct as contain
in the operand 36a + 2cl3 + dy), and reject the first term (or term independent o
9a. 9^, 9c, 9rf in the expansion) we have the required value of 4>({d?Sy)}. This value is
(9a4>9a + 9^4>96 + 9ya>9c)(36a + 2c/8 + *y);
performing the differentiations 9a, 9^, de, da, the value is
(3a9^ + 2/89y 4 79«) *,
i.e. we have * ({^y}) = {^«} (*)•
139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 229
15. Suppose now that 4> is a covariant of 8, then the operation <I> performed
upon any covariant of TJ gives rise to a covariant of the system
(f, 17, ...$a:, y, ...), (f, V» •••$^» y'» •••)» &c.
To prove this it is to be in the first instance noticed, that as regards (f, 17, ...$a?, y, ...), &c.
we have [aody] = 178^, &c. Hence considering [xdy], &c. as referring to the quantic U,
the operation 2 {aj0y} — aidy will be equivalent to [aidy] + 178^ — aSy, and therefore every
covariant of the s)rstem must be reduced to zero by each of the operations
iB = [xdy] + 179^  a^y.
This being the case, we have
ID . ^ = B* 4 ID (^),
^nations which it is obvious may be replaced by
^ind consequently (in virtue of the theorem) by
B.^ = B4> + ^3f(^),
a>.iD = ^B + {i73f}(4>);
^nd we have therefore
B . a>  4>.iD =  ({179^}  173^)(4>) ;
or, since 4> is a covariant of 0, we have iD . 4> = 4> . iD. And since every covariant
of the system is reduced to zero by the operation 39, and therefore by the operation
<>.19, such covariant will also be reduced to zero by the operation iD.<I>, or what is
the same thing, the covariant operated on by 4>, is reduced to zero by the operation
"9 and is therefore a covariant, i.e. <I> operating upon a covariant gives a covariant.
16. In the case of a quantic such as {7 =
(♦i^> y$^'» yOX
^e may instead of the new sets (f, ^), (f, rf)... employ the sets (y, — a?), (y', a?')» &c
The operative quantic 8 is in this case defined by the equation 0Cr=O, and if 4>
l)e, as before, any covariant of %, then 4> operating upon a covariant of U will give
« covariant of U. The proof is nearly the same as in the preceding case ; we have
instead of the equation ^({^y})= {179^} (*) the analogous equation
Avhere on the lefthand side [aody] refers to TJy but on the righthand side [acdy] refers
to 0, and instead of ID = {aSy} 4 ^^ — aSy we have simply ID = {aSy} — aSy .
230 AN INTRODUCTORY MEMOIR UPON QUANTICS. [139
17. I pass next to the quantic
which I shall in general consider under the form
(a, b,...b\ a^'^x, y)"»,
but sometimes under the form
(a, 6, ...6\ a'^x, y)**
the former notation denoting, it will be remembered,
and the latter notation
But in particular cases the coefficients will be represented all of them by unacceu
tuated letters, thus (a, 6, c, dP^x, y)* will be used to denote CM;* + 36a;^3ca:y* + (iy',
and (a, 6, c, d^x, yY will be used to denote cwc* + fta;^ + cjcy* f dy*, and so in all
similar cases.
Applying the general methods to the quantic
(a, 6, ...6\ a'$a?, y)"*,
we see that {y^*} = a96+ 269e...+m6'3a*,
in fact, with these meanings of the symbols the quantic is reduced to zero by each
of the operations {yS*} — y9«, {a;9y} — xdy ; hence according to the definition any function
which is reduced to zero by each of the lastmentioned operations is a covariant of
the quantic. But in accordance with a preceding remark, the covariant may be con
sidered as a rational and integral function, separately homogeneous in regard to the
&cients {x, y) and the coefficients (a, 6, ...&\ a^). If instead of the single set (a;, y)
the covariant contains the sets (^, y^), (^, y,), &c., then it must be reduced to zero
by each of the operations ly9»} — /Sy9aj, {aSy}— /Sady (where Sy3,B = yi3,B, fyiS^l ...), but
I shall principally attend to the case in which the covariant contains only the set
Suppose, for shortness, that the quantic is represented by U^ and let CTi, IT,,...
be what U becomes when the set (x, y) is successively replaced by the sets {xi^ y^),
(^i y%)i ^ Suppose moreover that 12 = 8^^8,^ — d^^dy,, &c., then the function
12P13«23^... CTjlTjCr,...,
in which, after the differentiations, the new sets {x^ yi), (a^, y^^... may be replaced
by the original set (x^ y), will be a covariant of the quantic {7. And if the number
39] AN INTRODUCTORY MEMOIR UPON QUANTIOS, 231
f differentiations be such as to make the facients disappear, ie. if the sum of all
he indices p, 9,... of the terms 12, &c. which contain the symbolic number 1, the
am of all the indices p, r, ... of the terms which contain the symbolic number 2,
nd so on, be severally equal to the degree of the quantic, we have an invariant,
lie operative quantic becomes in the case under consideration
he signs being alternately positive and negative; in fact it is easy to verify that this
xpression gives identically OCT^O, and any co variant of operating on a covariant
•f U gives rise to a covariant of U,
18. But the quantic
(a, 6, . . . b\ a'^x, y)*",
onsidered as decomposable into linear factors, ie. as expressible in the form
a(a?ay)(a?^y)...,
[ives rise to a fresh series of results. We have in this case
{y3*}= 3.4 dp...,
{xdy} =  (a + fi...) ada + a«. + /ff©^ + ... ;
n fact with these meanings of the symbc4s the quantic is reduced to zero by each
)f the operations {ady} — aSy, {y9«} — y3aj, and we have consequently the definition of
he covariant of a quantic considered as expressed in the form a(a? — ay)(a? — /3y),...
\jid it will be remembered that these and the former values of the symbols [xdy] and
ydg] are, when the same quantic is considered as represented under the two forms
a, 6, . . .i\ a'$a?, yy* and a (a? — ay) (a? — I3y). . . , identical
19. Consider now the expression
a« (^  ayy (a:  i8y)*...(a  i8)P... ,
¥here the sum of the indices j, p,.., of all the simple &ctors which contain a, the
lum of the indices k, p,... of all the simple factors which contain fi, Sec are respec
ively equal to the index of the coefficient a. The index d and the indices p, &c.
nay be considered as arbitrary, nevertheless within such limits as will give positive
ralues (0 inclusive) for the indices j, k,.,,.
The expression in question is reduced to zero by each of the operations
xdy] — xdy, [yds] — yd^ ; and this is of course also the case with the expressions
)btained by interchanging in any manner the roots a, fi, 7,..., and therefore with
the expression
a«2(a?ay)i(a?/3y)»...(a^)p...,
where 2 denotes a summation with respect to all the different permutations of the
roots a, fi, ... .
232 AN INTEODUCrrORY MEMOIR UPON QU ANTICS. [139
The function so obtained (which is of course a rational function of (a, 6, ...V, a))
will be a covariant, and if we suppose /i = md — 25/>, where Sp denotes the sum of all
the indices p of the different terms (a — fiy, &c., then the covariant will be of the
order fi (Le. of the degree fi in the facients x^ y), and of the degree in the co
efficienta
20. In connexion with this covariant
a* 2 (a? ~ ay)P (a;  i8y)*... (a  /3)P...,
of the order fi and of the degree in the coefHcients, of the quantic C/'=
a(a?~ay)(a?^y)...,
consider the covariant
of a quantic F=
in. which, after the differentiations, the sets (xi, yi), (x^, yj,), ... are replaced by the
original set (a?, y). The lastmentioned covariant will be of the order m (^ — ^) 4 /*,
and will be of the degree m in the coeflScients; and in particular if ^ = 5, i.e. if V
be a quantic of the order d, then the covariant will be of the order fA and of the
degree m in the coefficients. Hence to a covariant of the degree in the coefficients,
of a quantic of the order m, there corresponds a covariant of the degree m in the
coefficients, of a quantic of the order 0] the two covariants in question being each
of them of the same order /t. And it is proper to notice, that if we had commenced
with the covariant of the quantic F, a reverse process would have led to the
covariant of the quantic U. We may, therefore, say that the covariants of a given
order and of the degree in the coefficients, of a quantic of the order m, correspond
each to each with the covariants of the same order and of the degree m in the
coefficients, of a quantic of the order 0; and in particular the invariants of the degree
d of a quantic of the order m, correspond each to each with the invariants of the
degree ni of a quantic of the order 0. This is the law of reciprocity demonstrated
by M. Hermite, by a method which (I am inclined to think) is substantially identical
with that here made use of, although presented in a very different form: the dis
covery of the law, considered as a law relating to the number of invariants, is due
to Mr Sylvester. The precise meaning of the law, in the lastmentioned point of
view, requires some explanation. Suppose that we know all the really independent
invariants of a quantic of the order m, the law gives the number of invariants of
the degree m of a quantic of the order (it is convenient to assume > m), viz. of
the invariants of the degree in question, which are linearly independent, or asyzygetic,
Le. such that there do not exist any merely numerical multiples of these invariants
having the sum zero; but the invariants in question may and in general will be
connected inter se and with the other invariants of the quantic to which they belong
by nonlinear equations : and in particular the sjrstem of invariants of the degree m
will comprise all the invariants of that degree (if any) which are rational and integral
139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 233
functions of the invariants of lower degrees. The like observations apply to the system
of covariants of a given order and of the degree m in the coefficients, of a quantic
of the order ft
21. The number of the really independent covariants of a quantic (♦$«?, y)"* is
precisely equal to the order m of the quantic, Le. any covariant is a function
(generally an irrational function only expressible as the root of an equation) of any
m independent covariants, and in like manner the number of really independent in
variants is m — 2 ; we may, if we please, take m — 2 really independent invariants as
part of the system of the m independent covariants; the quantic itself may be taken
as one of the other two covariants, and any other covariant as the other of the two
covariants; we may therefore say that every covariant is a function (generally an
irrational function only expressible as the root of an equation) of m — 2 invariants, of
the quantic itself and of a given covariant.
22. Consider any covariant of the quantic
(a, 6, ... b\ a'Jx, yf,
and let this be of the order /i, and of the degree in the coefficients. It is very
easily shown that md — fi is necessarily even. In particular in the case of an invariant
(Le. when /t = 0) m0 is necessarily even^: so that a quantic of an odd order admits
only of invariants of an even degree. But there is an important distinction between
the cases of md^fi evenly even and oddly even. In the former case the covariant
remains unaltered by the substitution of (y, x), (a\ b\ ... b, a) for {x, y), (a, b, ... 6\ a');
in the latter case the effect of the substitution is to change the sign of the covariant.
The covariant may in the former case be called a symmetric covariant, and in the
latter case a skew covariant. It may be noticed in passing, that the simplest skew
invariant is M. Hermit e*s invariant of the degree 18 of a quantic of the order 5.
23. There is another very simple condition which is satisfied by every covariant
of the quantic
(a, b,...b\ a^x, y)*",
viz. if we consider the facients (x, y) as being respectively of the weights i, — i, and
the coefficients (a, 6, ...6\ a) as being respectively of the weights — ^m, — ^+1,
...^m — 1, ^m, then the weight of each term of the covariant will be zero. This is
the most elegant statement of the law, but to avoid negative quantities, the state
ment may be modified as follows: — if the facients (x, y) are considered as being of
the weights 1, respectively, and the coefficients (a, b...V, a') as being of the weights
0, 1, ...,m — 1, m respectively, then the weight of each term of the covariant will be
i(wi^ + /*)
^ I may remark that it was only M. Hermite^s important discovery of an invariant of the degree 18 of
a qnantio of the order 5, which removed an erroneous impression which I had been under from the oom
mencement of the subject, that mO was of necessity evenly even.
C. II. 80
234 AN INTRODUCTORY MEMOIR UPON QU ANTICS. [l39
24. The preceding laws as to the form of a covariant have been stated here by
way of anticipation, principally for the sake of the remark, that they so tso' define the
form of a covariant as to render it in very many cases practicable with a moderate
amount of labour to complete the investigations by means of the operation {ady} — xd^
and {ydg] — yd^^ In fact, for finding the covariants of a given order, and of a given
degree in the coefficients, we may form the most general function of the proper order
and degree in the coefficients, satisfying the prescribed conditions as to symmetry and
weight: such function, if reduced to zero by one of the operations in question, will,
on accoimt of the symmetry, be reduced to zero by the other of the operations in
question; it is therefore only necessary to effect upon it, e.g. the operation {ody} — a:9y,
and to determine if possible the indeterminate coefficients in such manner as to
render the result identically zero: of course when this cannot be done there is not
any covariant of the form in question. It is moreover proper to remark, as regards
invariants, that if an invariant be expanded in a series of ascending powers of the
first coefficient a, and the first term of the expansion is known, all the remaining
terms can be at once deduced by mere differentiations. There is one very important
case in which the value of such first term (i.e. the value of the invariant when a is
put equal to 0) can be deduced firom the corresponding invariant of a quantic of the
next inferior order; the case in question is that of the discriminant (or function
which equated to zero expresses the equality of a pair of roots); for by Joachimsthars
theorem, if in the discriminant of the quantic (a, b, ... 6\ a^$a?, y)^ we write a = 0, the
result contains 6* as a factor, and divested of this factor is precisely the discriminant
of the quantic of the order m — 1 obtained firom the given quantic by writing a =
and throwing out the &ctor x: this is in practice a very convenient method for the
calculation of the discriminants of quantics of successive orders. It is also to be
noticed as regards covariants, that when the first or last coefficient of any covariant
(i.e. the coefficient of the highest power of either of the facients) is known, all the
other coefficients can be deduced by mere differentiations.
Postscript added October 7th, 1854. — I have, since the preceding memoir was
written, found with respect to the covariants of a quantic (♦ $a7, y)"*, that a function
of any order and degree in the coefficients satisfying the necessary condition as to
weight, and such that it is reduced to zero by one of the operations {xdy} — xdy,
[ydxl—ydzf will of necessity be reduced to zero by the other of the two operations,
i.e. it will be a covariant; and I have been thereby led to the discovery of the law
for the number of asyzygetic covariants of a given order and degree in the coefficients;
from this law I deduce as a corollary, the law of reciprocity of MM. Sylvester and
Hermite. I hope to return to the subject in a subsequent memoir.
140]
235
140.
RESEARCHES ON THE PARTITION OF NUMBERS.
[From the Philosophical Transactions of the Royal Society of London, vol. cxlv for the
year 1855, pp. 127—140. Received April 14,— Read May 24, 1855.]
I PROPOSE to discuss the following problem : " To find in how many ways a
number q can be made up of the elements a, 6, c, ... each element being repeatable
an indefinite number of times." The required number of partitions is represented by
the notation
P(a, 6, c, ...)?,
and we have, as is well known,
P{a,b.c, ...)3 = coefficient afl in ( i_^)(i _^)(i _^)... .
where the expansion is to be effected in ascending powers of x.
It may be as well to remark that each element is to be considered as a separate
and distinct element, notwithstanding any equalities which may exist between the
numbers a, 6, c, . . . ; thus, although a = 6, yet a f a + a h &c. and a 4 a f 6 f &c. are to
be considered as two different partitions of the number q, and so in all similar cases.
The solution of the problem is thus seen to depend upon the theory, to which I
now proceed, of the expansion of algebraical firactions.
Consider an algebraical fraction v ,
jx
where the denominator is the product of any number of factors (the same or different)
of the form 1 — a^**. Suppose in general that [1 — a^] denotes the irreducible factor of
1 — af^, L e. the factor which, equated to zero, gives the prime roots of the equation
1  a:^ = 0. We have
30—2
236 RE8EARCHBS ON THE PARTITION OF NUMBEB& [l40
where m' denotes any divisor whatever of m (unity and the number m itself not
excluded). Hence, if a represent a divisor of one or more of the indices m, and h
be the number of the indices of which a is a divisor, we have
/a? «n[l «•]*.
Now considering apart from the others one of the multiple £EU^rs [! — «•]*, we
may write /a: = [1  «•]*/«•
^ is decomposed
4 Ac,
where I{x) denotes the integral part, and the &c. refers to the fractional terms
depending upon the other multiple factors such as [1 — ««]* The functions Qx are
to be considered as functions with indeterminate coefficients, the degree of each such
frmction being inferior by unity to that of the corresponding denominator; and it is
proper to remark that the number of the indeterminate coefficients in all the frmctions
6x together is equal to the degree of the denominator fx,
dx
The term (aS,)*"' rfT^ ™*y ^ reduced to the form
qx g,x
[i "^ "^ [1  a^]*> "^ ^^'^
the functions gx being of the same degree as Ox, and the coefficients of these functions
being linearly connected with those of the function 0x, The first of the foregoing
terms is the only term on the righthand side which contains the denominator [1 — a;"]* ;
hence, multiplying by this denominator and then writing [1 — a;*] = 0, we find
ihx
which is true when x is any root whatever of the equation [1 — a;"] = 0. Now by
means of the equation [1 — a;^] = 0, j~ may be expressed in the form of a rational and
integral function Ox, the degree of which is less by unity than that of [1— a^]. We
have therefore Gx=^gx, an equation which is satisfied by each root of [1— a^] = 0,
and which is therefore an identical equation ; gx is thus determined, and the coefficients
of Ox being linear functions of those of gx, the function Ox may be considered as
determined And this being so, the function
fx ^" [laf^]
140] BESEAKCHES ON THE PARTITION OF NUMBERS. 237
will be a fraction the denominator of which does not contain any power of [1 — af^]
higher than [1 — «*]*"* ; and therefore 0iX can be found in the same way as 0x, and
similarly OtX, and so on. And the fractional parts being determined, the integral part
may be found by subtracting from ^ the sum of the fractional parts, so that the fraction
J can by a direct process be decomposed in the abovementioned form.
Particular terms in the decomposition of certain fractions may be obtained with
great facility. Thus m being a prime number, assume
1 St ^ ^^
= &c. f
then observing that (1 — aJ^) = (l — ip)[l — a?^], we have for [1— a?^] = 0,
5a; =
Now u being any quantity whatever and x being a root of [1 — x^"] = 0, we have
identically
[1  u'**] =(t4  x)(ua?) ... (m  3^^) ;
and therefore putting ti = l, we have m = (l— a?) (1 — «•)... (1— a;*^*), and therefore
5a;=,
m
whence
1 _o 1 1
Again, m being as before a prime number, assume
= &c. 4
{lx){\a?)...{laf^) ' [la;"']'
we have in this case for [1 — af^] = 0,
0x^
which is immediately reduced to 5a? = — , . Now
'' ml—x
tt — a? u — x ^ ^
or putting u » 1,
= (TOl)+(m2)a;...+a;«»;
lx
238 RESEARCHES ON THE PARTITION OF NUMBERS. [l40
aud substituting this in the value of Oxy we find
1 « J_ (m  1) + (tw  2)a? ... + af^
(la:)(la;«)...(la?'~)'" ■*"m« [laJ~]
The preceding decomposition of the fraction ? gives very readily the expansion ot
the fraction in ascending powers of x. For, consider a fiuction such as
0x
where the degree of the numerator is in general less by unity than that of the
denominator ; we have
la;« = [la:«]n[la:«'],
. where a' denotes any divisor of a (including unity, but not including the number a
itself). The fraction may therefore be written under the form
0xTl [1  Qf"']
where the degree of the numerator is in general less by unity than that of the
denominator, Le. is equal to a — 1. Suppose that b is any divisor of a (including
unity, but not including the number a itself), then 1 — a:* is a divisor of H [1 — sfi^, and
therefore of the numerator of the fraction. Hence representing this numerator by
A^ + A^x ... + AaiOf^',
and putting a = 5c, we have (corresponding to the case 6 = 1)
^0 "H "^i + A^ ... + A^^i ^ 0,
and generally for the divisor 6,
Ai + Aif^i . . . + 4 (e_n6+i = 0,
*
^6—1 + <4s6~i . . . + ^efri = 0.
Suppose now that a^ denotes a circulating element to the period a, Le. write
a, = 1, 9 = (mod. a),
a, = in everj' other case;
a frmction such as
A^ + AxQ^\ . . . + Af^ia^^^^i
will bo a circulating ftuiction, or circulator to the period a, and may be represented
by the notation
(A^y ill, •••'^•^ circlor o^.
140] RESEARCHES ON THE PARTITION 01^ NUMBERS. 239
In the case however where the coefficients A satisfy, for each divisor b of the number
a, the abovementioned equations, the circulating function is what I call a prime
circulator, and I represent it by the notation
(^0, ^i, ,,, ^(g^i) per Ciq.
By means of this notation we have at once
0x
coefficient afl in tt — ^1= (^o, Ai...Aa^i) per aq,
and thence also
0x
coefficient a?^ in (aidxY n — ;;ii ~ 5''^('^o» ^i^^ai) P^r aq.
Hence assuming that in the fraction ^ the degree of the numerator is less than that
of the denominator (so that there is not any integral part), we have
coefficient a^ in ^ = X q^(Ao, A^, ..,Aa^i) per a^;
or, if we wish to put in evidence the noncirculating part arising from the divisor a = 1,
coefficient .^^ in ^ = A^^ + B^ ...^Lq^M
+ 2 /(A, A^...Aa^) per a,;
where k denotes the number of the factors 1 — a?"* in the denominator fx, a is any
divisor (unity excluded) of one or more of the indices m; and for each value of a
r extends from r = to r = & — 1, where k denotes the number of indices m of which
a is a divisor. The particular results previously obtained show, that m being a
prime number,
coefficient a^ in (i ,^)(i ^^) ^,, (i ,^^ = &c.4^( 1,1, 0, 0, ...) F^ ^g>
and
coefficient ^ in (i _^)(i _^)...(t _^„) = &c. + ^,(ml,l,l. ...) perm,.
Suppose, as before, that the degree of ^ is less than that of fx, and let the
analytical expression above obtained for the coefficient of a;? in the expansion in
^)X
ascending powers of x of the fraction ^ be represented by Fq, it is very remarkable
(hx
that if we expand ^ in descending powers of a?, then the coefficient of afl in this
new expansion {q is here of course negative, since the expansion contains only
negative powers of x) is precisely equal to —Fq; this is in fact at once seen to be
240 RESEARCHES ON THE PARTITION OF NUMBERS. [140
the case with respect to each of the partial fractions into which ^ has been de
composed, and it is consequently the case with respect to the fraction itself \ This
gives rise to a result of some importance. Suppose that ^ and fx are respectively
of the degrees J!V and 2); it is clear from the form of ^a? that we have /() = (— /a?"^;
and I suppose that i>x is also such that ^ () =(±)'ic"'^^; then writing D^N^h,
and supposing that ^ is expanded in descending powers of x, so that the coefficient
of ^ in the expansion is ^Fq, it is in the first place clear that the expansion will
commence with the term ar~*, and we must therefore have
Fq^O
for all values of q from } = — 1 to 9 = — (A — 1).
Consider next the coefficient of a term ic~*~«, where 9 is or positive; the
coefficient in question, the value of which is — ^(— A — 5), is obviously equal to the
coefficient of a^+« in the expansion in ascending powers of x of — ~, Le. to
(±)V/ coefficient a^+« in ^,
. fx
or what is the same thing, to
(±y(y coefficient 0^ in ^;
jx
and we have therefore, q being zero or positive,
F(hq) = (±nyFq.
In particular, when ^ = 1, Fq^O
for all values of q from j = — 1 to 5 = — (D — 1) ; and q being or positive,
F(^D^q)^{^r'Fq.
The preceding investigations show the general form of the ftmction P(a, 6, c,...)?,
viz. that
P(a, b, c,...)3 = 4g*~^ + 5g'*»...+Z5 + If + 29^(^10, Ai,...Ai^i) per Ig,
a formula in which k denotes the number of the elements a, b, c, ...^c., and I is
any divisor (unity excluded) of one or more of these elements; the summation in the
case of each such divisor extends from r = to r = A? — 1, where k is the number of
the elements a, b, c, ...&c. of which Z is a divisor; and the investigations indicate
^ The property is a fondamental one in the general theory of deyelopments.
140] RESEARCHES ON THE PARTITION OF NUMBERS. 241
how the values of the coefficients A of the prime circulators are to be obtained. It
has been moreover in eflfect shown, that ifi) = a + 6 + c + ..., then, writing for shortness
P(q) instead of P(a, b, c, ...)?> ^^ have
P(q) =
for all values of q from } = — 1 to 5^ = — (D — 1), and that q being or positive,
P(Dq) = ir'P(q);
these last theorems are however uninterpretable in the theory of partitions, and the}'
apply only to the analytical expression for P(q).
I have calculated the following particular results: —
P(l. 2)y =i{2? + 3
+ (1, 1) per 2,1
P(l, 2, 3)9 =i^J6g» + 36? + 47
72
(.
+ 9(1, 1) per 2g
+ 8(2, 1, 1) per 3jl
P(l, 2, 3. 4)9 =^29» + 309' + 1353 + 175
+ (9? + 45)(l, 1) per 2,
+ 32 (1, 0,  1) per 3,
+ 36 (1, 0, 1, 0) per 4,1
P(l, 2, 3, 4, 5) g = gg^ JSO g« + 900 g* + 9300 5' + 38250 g + 50651
+ (13503 + 10126) (1, 1) per 2,
+ 3200 (2,  1,  1) per 3,
+ 5400 (1, 1,  1,  1) per 4,
+ 3466 (4, 1,1,  1,  1) per 5,
P(2)q =\[l
+ (1,  1) per 2,
^(2.3)9 =^{29 + 5
+ 3 (1,  1) per 2,
+ 4(1, 1,0) per 3, 1
C. II.
31
242
RESEARCHES ON THE PARTITION OF NUMBERS.
[140
!"{% 3, 4)g
288
P(2, 3,4. 5)5 =
P(2, 3, 4, 5, 6)3 =
6 ^r' + 54 5 + 107
+ (18g + 81)(l, 1) per 2,
+ 32 (2,  1,  1) per 8,
+ 36 (1, 1, 1,1) per 4,1
+ (45} + 315)(l, 1) per 2,
+ 160 (1,  1. 0) per 3,
+ 180 (1, 0,  1, 0) per 4,
+ 288 (1,  1, 0, 0, 0) per 3,1
[lO 3« + 4009» + 5550 3»+ 31000 q + 56877
+ (450 3»+ 9000? + 39075) (1, 1) per 2,
(1,  1. 0) per 3,
(21,  19,  2) per 3,
(1, 0,  1, 0) per 4,
(4,  1,  1,  1,  1) per 5,
(1, 1,2,1, 1. 2) per 6,1
172800
+ 3200 9
+ 1600
+ 10800
+ 6912
+ 4800
Pil, 2, 3, o)q =
JL_ (
720
P(\, 2, 2, 3, 4)9 =
4y' + 669"+ 3249 + 451
+ 45 (1,  1) per 2,
+ 80 (1,  1, 0) per 3,
+ 144(1, 0, 0, 0, 1) per 5,1
g^ je 9* + 144 9" + 1194 3» + 3960 } + 4267
+ (64 9" + 648 9 + 1701) (1, 1) per 2,
+ 256 (2, 1,1) per 3,
+ 432 (1, 0, 1, 0) per 4,1
P(8)9
^>
+ 1 (1, 1) per 2,
+ 2 (1, 0,  1, 0) per 4,
+ 8(1, 0, 0, 0, 1, 0, 0, 0) per 8,1
140] RESEABCHES ON THE PARTITION OF NUMBERS. 243
+ 7 (1,  1) per 2,
+ 14 (1. 1,1, 1) per 4,
which are. I think, worth preserving.
+ 16 (3, 2, 1, 0, 1,2,3) per 7^
+ 66 (0, 1,1, 0, 0, 1, 1, 0) per sj,
Received April 14,— Read May 3 and 10, 1855.
I proceed to discuss the following problem: "To find in how many ways a
number q can be made up as a sum of m terms with the elements 0, 1, 2, . . . A;,
each element being repeatable an indefinite number of timea" The required number
of partitions is represented by
P(0, 1, %...k)^q,
and the number of partitions of q less the number of partitions of j — 1 is repre
sented by
F(0, 1, 2, ...&)«}.
We have, as is well known,
P(0. 1. 2....A)»3 = coeffieient ^^ in (i _,)(i _i)...(i _^,) .
where the expansion is to be effected in ascending powers of z. Now
1 i.lz^' (la^^0(la^+«) ,
the general term being
(1  a^+0(l  a:*+») ... (1  a^^"*)
(la:)(la;»)... {laf^)
or, what is the same thing,
(la:)(la;«)... (1a?*)
and consequently
P(0. 1. 2. ...*).5 = eoeflBeient ^ in ^' (il^d C)..r(i S) '
to transform this expression I make use of the equation
(l+a:^)(l+.;»^)...(l+.*^) = l+L__^+ (\,^/(\^^) V + &C.,
31—2
z^,
s^.
244 RESEARCHES ON THE PARTITION OF NUMBERS. [140
where the general term is
and the series is a finite one, the last term being that corresponding to s^k, viz.
^(i+i)^ Writing —a^ for z, and substituting the resulting value of
(I  aH^i) (1  r~+«) . . . (1  a»+*)
in the formula for P(0, 1, 2, ...4)"y, we have
P(0. 1. 2. ... A)v = S,^(y coeffideot ^ in (i.^)(i_^)...(i_^)(l^)(l<'^).(lx*n '
where the summation extends firom ^^0 to s^k; but if for any value of s between
these limits «m + ^(«+l) becomes greater than q, then it is clear that the summation
need only be extended firom 8=^0 to the last preceding value of s, or what is the
same thing, firom « = to the greatest value of s for which j — «m — ^«(«+l) is
positive or zenx
It is obvious, that if y > km, then
P(0, 1, 2...A)}«0;
and moreover, that if ^ > Ikm, then
P(0, 1, 2,...it)~^«P(0, 1, 2,...it)~.itmtf,
so that we may always suppose j > Jibm. I write therefore q = ^(kni'a) where a is
xero or a {positive integer not greater than km, and is even or odd according as Arm
18 oven or inld. Substituting this value of q and making a slight change in the
form of the n^sult, we have
whon« tht^ Hummation extends firom 9 = to the greatest value of s for which
(U* «)'H  ^ot i«(^+ 1) IB positive or zero. But we may, if we please, consider the
miiuiimtion m extending, when k is even, fix)m 8^0 to 8^\k — \, and when k is odd,
\\\\\\\ /iO to j» "• i (il' 1); the terms corresponding to values of 8 greater than the
gh»Hti»Hi value for which (iifc«) m Ja J«(«4 1) is positive or zero being of course
equal to X(«rt»« It may be noticed, that the firaction will be a proper one if
ri* (k H)(kH'k\)\ or substituting for 8 its greatest value, the firaction will be a
iiiniirr Olio for all values of «, if, when it is even, o< Jifc(A: + 2), and when k is odd,
«• \(k\ \)(k \ HV
Wo have in a nimilar maimer,
/••((). I. 2. ..Arv = coefficient «•,« in (r ^)(l^X(l ^*^) '
140] RESEABCHE8 ON THE PARTITION OF N(JMB£R8. 245
which leads to
^(0, 1, 2...ik)~i(Jtma) =
where the summation extends, as in the former case, firom « = to the greatest value
of «, for which (Jifc — «)m — ^a — i«(« + 1) is positive or zero, or, if we please, when k is
even, from « = to « = iA?— 1, and when 8 is odd, from « = to « = J(A — 1). The
condition, in order that the fruction may be a proper one for all values of s, is,
when k is even, a + l<\k(k + 2), and when k is odd, a + 1 < i(A + l)(A: + 3).
To transform the preceding expressions, I write when k is odd jc^ instead of x,
and I put for shortness instead of ^k — s or 2(iA: — «), and y instead of Jah J«(«»l)
or a + « (« + 1) ; we have to consider an expression of the form
coefficient a^ in =r ,
where Fx is the product of factors of the form 1 — «:*. Suppose that a is the least
common multiple of a and 0, then (1 — a*') h (1 — «*) is an integral function of x,
equal x^ suppose, and 1 ? (1 — a;*) = ^^ s (1  a?*'). Making this change in all the
factors of Fx which require it (Le. in all the factors except those in which a is a
multiple of 0), the general term becomes
coefficient ar^ in —tz — ,
where Go; is a product of factors of the form 1 — af^'y in which a' is a multiple of 0,
i.e. Go; is a rational and integral fiinction of of. But in the numerator a!*Hx we may
reject, as not contributing to the formation of the coefficient of a^, all the terms in
which the indices are not multiples of 0\ the numerator is thus reduced to a rational
and integral function of afy and the general term is therefore of the form
coefficient af^ in Wv*
or what is the same thing, of the form
coefficient af* m — ,
icx
where xx is the product of factors of the form 1 — ic*, and Xo; is a rational and integral
function of x. The particular value of the fraction depends on the value of s: and
uniting the dififerent terms, we have an expression
\x
coefficient x^ in 8m (—V — ,
^ KX
^hich is equivalent to
coefficient a"* in ^ ,
fx
246 RESEARCHES ON THE PARTITION OF NUMBERS. [l40
where /i; is a product of factors of the form 1 af^, and ^ is a rational and integral
function of x. And it is clear that the fraction will be a proper one when each
of the fractions in the original expression is a proper Auction, i.e. in the case of
P(0, 1, 2...Jk)'»J(*^«)» when for k even, a<iik(A:+2), and for k odd, a<i(Jfc+l)(it+3);
and in the case of P'{0, 1, 2 ... A:)^ J(A?m — a), when for k even, a+l<\k(k + 2)y and
for k odd, a+1 <l{k\l){khS).
We see, therefore, that
P(0, 1, 2...ifc)«»K*wiaX
and
1^(0, 1, 2...ifc)«i(^«X
are each of them of the form
coefficient af* in ?,
where fx is the product of factors of the form 1 — a*, and up to certain limiting values
(dX
of a the fiuction is a proper fraction. When the fraction ~ is known, we may there
fore obtain by the method employed in the former part of this Memoir, anal3rtical
expressions (involving prime circulators) for the functions P and P',
As an example, take
P(0, 1, 2, 3)'^m,
which is equal to
1
coefficient a^ in
— coefficient af* in
(la;«)(la?*)(l««)
(laj«)(lic»)(l «?•)•
The multiplier for the first fraction is
which is equal to
1+ a;* h ar* + ic« + 2a;» + a^' + «*'.
Hence, rejecting in the numerator the terms the indices of which are not divisible
by 3, the first term becomes
coefficient ac^ in
(la^)(laj»«)(la^)'
or what is the same thing, the first term is
1+a^ + a?*
coefficient xl^ in
(la^)»(la?*)'
140] RESEARCHES ON THE PARTITION OF NUidBERS. 247
and, the second term being
— coefficient aS^ in
(la:»)«(la?*)'
1 +ip*
we have P(0, 1, 2, 3)'"w = coeflBcient a^ in /i ^a^vn ar<V
and similarly it may be shown, that
P(0, 1, 2, 3)"* i(3w  1) = coefficient af» in
(lic«)»(lar*)
As another example, take
P'(0, 1, 2, 3, 4, 5)fm,
which is equal to
1
coefficient a*^ in
— coefficient a^ in
+ coefficient off^ in
(1  a?*) (1  a^) (1  ic*) (1  a?io)
(^
(1  a;«) (1  «?•) (1  a^) (1  a:«)
(1  a;«) (1  a?*) (1  a;*) (1  ««) •
The multiplier for the first firaction is
which is a function of a? of the order 36, the coefficients of which are
1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 4, 4, 6, 4, 6, 5, 7, 5, 7, 5, 7, 5, 6, 4, 6, 4, 4, 3, 4. 2, 3, 1, 2, 1, 1, 0, 1,
and the first part becomes therefore
coefficient aS^ m — p. ^rr — r— — —^— — — — .
(1 a^)(l a^)(l aj'Xl aJ®)
The multiplier for the second fraction is
(la^)(la^')(la;»*)
(la;»)(laj*)(l a^)'
which is a function of a? of the order 14, the coefficients of which are
1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1 ;
and the second term becomes
iX3 L «• 2a^ + 2a:* + 3a^ + «" + a:"
coefficient a^ m (i ^^y^x ar^)(l a:") >
248 RESEABCHES ON THE PARTITION OF NUMBERS. [UO
and the third term is
coefficient «** in
Now the fractions may be reduced to a common denominator
(1 a;«)(l a?*)(l «•)(! «!»)
1 — a^
by multipljring the terms of the second fraction by :. — — (= 1 + a;* + a?*), and the terras
1 — «"
of the third Auction by ^ ^ (= 1 + a?*) ; performing the operations and adding, the
numerator and denominator of the resulting fraction will each of them contain the
factor 1 — a^ ; and casting this out, we find
P(0, 1, 2, 3, 4, o)** fm = coefficient ai^ in
(la?*)(la;«)(la:»)'
I have calculated by this method several other particular cases, which are given
in my "Second Memoir upon Quantics", [141], the present researches were in fact
made for the sake of their application to that theory.
Received April 20,— Read May 3 and 10, 1855.
Since the preceding portions of the present Memoir were written, Mr Sylvester
has communicated to me a remarkable theorem which has led me to the following
additional investigations ^
Let ^ be a rational fraction, and let (a? — a?i)* be a &ctor of the denominator fr,
/«
then if
denote the portion which is made up of the simple fractions having powers of x — Xi
for their denominators, we have by a known theorem
. ^l = coefficient  in %j — : — ( .
Now by a theorem of Jacobi's and Cauchy's,
coefficient  in ^f: = coefficient  in F(yyt)'^'t]
Z V
whence, writing Xi + z=^ ^«~S we have
J^l = coefficient  in i ^; ^ .
(/a:],, t x^x^f(xi€r*)
1 Mr Sylvester's researohes are published in the Quarterly Mathematical Journal^ July 1855, [vol. i. pp.
141—152], and he has there given the general formula as well for the eiroulating as the nondronlating part
of the expression for the number of partitions.— Added 28rd February, 1856.— A. G.
140] RESEARCHES ON THE PARTITION OF NUMBERS. 249
Now putting for a moment x=^x^(f, we have
1 1 1^1
+ ^tf — 77 T\ + • • • »
x^x^ a:i(l6'+*) iri(l6') ' "^ x^{\^)
and 3^ = ar9jr, whence
Xi^xtf Xi — x 1 Xi — x 1.2^ Xi — x
the general term of which is
fSl
Hence representing the general term of
Xi<f> (XifT*)
by x^i<~*, 80 that
X^i = coefficient ^ m ^' f(x.e*) '
we find, writing down only the general term,
i/4*r *'* "^ n(* 1) ("^^^U^"^ ••• '
where the value of x^ depends upon that of «, and where 8 extends from « = 1 to « = Ar.
Suppose now that the denominator is made up of £sictors (the same or different)
of the form 1 — a?"*. And let a be any divisor of one or more of the indices m,
and let k be the number of the indices of which a is a divisor. The denominator
contains the divisor [1 — af^f, and consequently if p be any root of the equation
[!—«"] = 0, the denominator contains the fetctor (p — a?)*. Hence writing p for Xi and
taking the sum with respect to all the roots of the equation [1 — ic*] = 0, we find
vhere vp = coefficient  in ^* jv^^v >
and as before 8 extends from ^ = 1 to 8==k. We have thus the actual value of the
function $x made use of in the memoir.
A preceding formula gives
coefficient t in S^\ f
^Hich is a very simple expression for the noncirculating part of the fraction ^
^*^is is, in fact, Mr Sylvester's theorem above referred to.
c. n. 32
250
[141
141.
A SECOND MEMOIR UPON QUANTICS.
[From the Philosophical Transactiona of the Royal Society of London, vol. CXLVI. for the
year, 1856, pp. 101—126. Received April 14,— Read May 24, 1855.]
The present memoir is intended as a continuation of my Introductory Memoir
upon Quantics, t. CXLIV. (1854), p. 245, and must be read in connexion with it ; the
paragraphs of the two Memoirs are numbered continuously. The special subject of
the present memoir is the theorem referred to in the Postscript to the Introductory
Memoir, and the various developments arising thereout in relation to the number and
form of the covariants of a binary quantic.
25. I have already spoken of asyzygetic covariants and invariants, and I shall have
occasion to speak of irreducible covariants and invariants. Considering in general a
function u determined like a covariant or invariant by means of a system of partial
differential equations, it will be convenient to explain what is meant by an asyzygetic
integral and by an irreducible integral. Attending for greater simplicity only to a
single set (a, 6, c, . . .), which in the case of the covariants or invariants of a single
function will be as before the coefficients or elements of the function, it is assumed
that the system admits of integrals of the form u^ P, u ^ Q, &c., or as we may
express it, of integrals P, Q, &c., where P, Q, &c. are rational and integral homogeneous
functions of the set (a, 6, c, ...), and moreover that the system is such that P, Q, &c.
being integrals, <f>(P, Q, •*•) is also an integral Then considering only the integrals
which are rational and integral homogeneous functions of the set (a, b, c, ...), integrals
P, Q, JB, ... not connected by any linear equation or syzygy (such as \P + fiQ + vR ... 0),Q)
are said to be asyzygetic; but in speaking of the asyzygetic integrals of a particular
degree, it is implied that the integrals are a system such that every other integral of
1 It is hardly necessary to remark, that the multipliers X, fi, r, ... , and generally any eoefficients or
quantities not expressly stated to contain the set (a, b, c, ...), are considered as independent of the set, or
to use a conyenient word, are considered as "trivials."
141] A SECOND MEMOIR UPON QU ANTICS. 251
the same degree can be expressed as a linear function (such as XP + /tQ + ]/i2...) of
these integrals; and any integral P not expressible as a rational and integral homo
geneous fnnction of integrals of inferior degrees is said to be an irreducible integral
26. Suppose now that A^^ A^, A^, &c. denote the number of asyzygetic integrals
of the degrees 1, 2, 3, &c. respectively, and let a^, a,, a,, &c. be determined by the
equations
A = iai(ffi + l) + or„
^8 = i «! («! + l)(ai + 2) + aia,hor3,
44 = ^ai(ai+l)(«i+2)(ai + 3)iai(orihl)a>laia, + ia,(a,+ l)ha4, &c.,
or what is the same thing, suppose that
. l + A^x + A^\&c. =(la:)'^*(la;«)"^(la:»)"^...;
a little consideration will show that a^, represents the number of irreducible integrals
of the degree r less the number of linear relations or syzygies between the composite
or nonirreducible integrals of the same degree. In £sict the asyzygetic integrals of
the degree 1 are necessarily irreducible, i.e. Ai^a^. Represent for a moment the
irreducible integrals of the degree 1 by X, X\ &c., then the composite integrals
Z*, XX\ &c., the number of which is iai(ai + l), must be included among the asyzygetic
integrals of the degree 2; and if the composite integrals in question were asyzygetic,
there would remain ilj — i «!(«! + 1) for the number of irreducible integrals of the
degree 2 ; but if there exist syzygies between the composite integrals in question, the
number to be subtracted from A^ will be ^^^(ai + l) less the number of these syzygies,
and we shall have ^a — i ai(ai + l), ie. ffj equal to the number of the irreducible
integrals of the degree 2 less the number of syzygies between the composite integrals
of the same degree. Again, suppose that Oj is negative = — A, we may for simplicity
suppose that there are no irreducible integrals of the degree 2, but that the com
posite integrals of this degree, X^, XX', &c., are connected by fi^ syzygies, such as
\X* + fiXX' + &C. = 0, XiZ* + fiiXX' + &c. = 0. The asyzygetic integrals of the degree 4
include X\ X*X', Sec, the number of which is ^ Oi (ce, + 1) (Oi + 2) (a^ + 3) ; but these
composite integrals are not asyzygetic, they are connected by syzygies which are
angmentatives of the fi^ syzygies of the second degree, viz. by syzygies such as
(XX« + fiZZ'...)Z» = 0, (XZ»h/iZZ'...)ZZ' = 0, &c. (XiZ«+/AiZZ'...)Z« = 0,
(XiZ» + /AiZZ'...)ZZ' = 0, &c.,
the number of which is iai(aihl))8j. And these syzygies are themselves not asyzygetic,
they are connected by secondary syzygies such as
Xj(VZ« + mX'Z'...)Z» + /Ii(XZ« + /iaZZ'...)ZZ' + &c.
X(XiZ>+/iiZZ'...)Z»fi(XiZ« + AhZZ'...)ZZ'&c. = 0, &c. &c.,
32—2
252 A SECOND MEMOm UPON QUANTIGS. [141
the number of which is i)9s09, — 1). The real number of syzygies between the com
posite integrals X\ X*X\ &c. (Le. of the syzygies arising out of the Pt syzygies
between X*, XX\ &c.), is therefore i ai(ai + 1)A — ii8,()8j— 1), and the number of
integrals of the degree 4. arising out of the integrals and syzygies of the degrees
1 and 2 respectively, is therefore
or writing —a, instead of )8,, the number in question is
A «!(«! + l)(or, + 2)(ai h 3) + i ai(a, + 1) or, + i a, (a, + 1).
The integrals of the degrees 1 and 3 give rise to ttiO, integrals of the degree 4; and if
all the composite integnJs obtained as above were asjrzygetic, we should have
^4Affi(ai + l)(ai + 2)(aj + 3)iai(ar,hl)a,ia,(a,hl)aia,,
i.e. OL^ as the number of irreducible integrals of the degree 4; but if there exist any
further syzygies between the composite integrals, then a^ will be the number of the
irreducible integrals of the degree 4 less the number of such further syzygies, and the
like reasoning is in all cases applicable.
27. It may be remarked, that for any given partial differential equation, or system
of such equations, there will be always a finite number v such that given v independent
integrals every other integral is a function (in general an irrational fimction only
expressible as the root of an equation) of the v independent integrals; and if to these
integrals we join a single other integral not a rational function of the v integrals, it is
easy to see that every other integral will be a rational function of the y + 1 integrals :
but every such other integral will not in general be a rational and integral fimction of
the ]/ + 1 integrals ; and lincorrecf] there is not in general any finite number whatever
of integrals, such that every other integral is a rational and integral function of these
integrals, Le. the number of irreducible integrals is in general infinite ; and it would seem
that this is in fact the case in the theory of covarianta
28. In the case of the co variants, or the invariants of a binary quantic, A^ \a given
(this will appear in the sequel) as the coefficient of o;^ in the development, in ascending
powers of a?, of a rational fi:uction ^ , where yi is of the form
(1  a:)^(l  «•)^ . .(1  a:*y»,
and the degree of ^ is less than that of ^. We have therefore
and consequently
<^=(1 a?)*Xlaj«y*"...(l af'Y'^O a*+')"**^' •
141] A SECOND MEMOIR UPON QU ANTICS. 253
Now every rational factor of a binomial l—af^iB the irreducible factor of 1 — af^',
where m is equal to or a submultiple of m. Hence in order that the series a^, or,, 3s»*
may terminate, ^ must be made up of factors each of which is the irreducible factor
of a binomial 1 — af^, or if ^ be itself irreducible, then <f>x must be the irreducible
factor of a binomial 1 — a^. Conversely, if ^ be not of the form in question, the
series a^, a,, a^, &c. will go on ad infinitum, and it is easy to see that there is no point
in the series such that the terms beyond that point are all of them negative, i.e. there
will be irreducible covariants or invariants of indefinitely high degrees; and the number
of covariants or invariants will be infinite. The number . of invariants is first infinite in
the case of a quantic of the seventh order, or septimic ; the number of covariants is first
infinite in the case of a quantic of the fifth order, or quintic. [As is now well known,
these conclusions are incorrect, the number of irreducible covariants or invariants is
in every case finite.]
'2d. Resuming the theory of binary quantics, I consider the quantic
(a, 6,...6\ a^$a?, y)*".
Here writing
[ady] = mbda + (m — 1 ) cS^. . .+ a'9y, = )',
any function which is reduced to zero by each of the operations X — yd^, Y—xdy is a
CO variant of the quantic. But a co variant will always be considered as a rational
and integral function separately homogeneous in regard to the facients {x, y) and to
the coeflBcients (a, 6,...6\ d). And the words order and degree will be taken to refer
to the facients and to the coeflBcients respectively.
I commence by proving the theorem enunciated. No. 23. It follows at once from
the definition, that the covariant is reduced to zero by the operation
X — ydx . Y — xdy — Y— xdy . X — yd^t
which is equivalent to
X . F— Y ,X+ydy — aSx'
Now
X.Y = XY+X(Y)
Y.X=YX+Y(X\
where XY and YX are equivalent operations, and
X(F)= lmada^2(ml)bdt...+ mlb'dt',
Y(X)= wl6aft...+ 2(m ])b'dt^\„ia:da,
whence
Jf (F) F(Jf) = 7yia9a + (m — 2)63ft...(m2)6'9e> ma'3a, =* suppose,
and the covariant is therefore reduced to zero by the operation
k + ydy — a^x
Now as regards a term a*b^..M^d''\a^y^, we have
fc = wa + (i?i2)/8..., '(m2)/3"md
ydyaSx^'ji;
254 A SECOND MEMOIR UPON QUANTIC8. [141
and we see at once that for each term of the covariant we must have
ma + (m  2) /8... (m  2) ;8^  tna' + j  1 = 0,
i.e. if (x, y) are considered as being of the weights ^, —^ respectively, and (a, b,...b\ a)
as being of the weights — fn, — ^m + 1, ...^m— 1, ^m respectively, then the weight
of each term of the covariant is zero.
But if {x, y) are considered as being of the weights 1, respectively, and (a, 6,...6\ a)
as being of the weights 0» l,...m — 1, m respectively, then writing the equation under
the form
and supposing that the covariant is of the order ^ and of the degree 0, each term of
the covariant will be of the weight \ (mO + /i).
I shall in the sequel consider the weight as reckoned in the lastmentioned manner.
It is convenient to remark, that as regards any function of the coefficients of the degree
6 and of the weight 9, we have
X.YY.X^mdiq.
30. Consider now a covariant
{A, B,...B, A^\x, yY
of the order n and of the degree 0\ the covariant is reduced to zero by each of the
operations X—yd,, T — aldy, and we are thus led to the systems of equations
XA^O,
XB = ijlA,
XC = (jjLl)B.
Xff= 2(7.
XA'=ff ;
and
YA = B,
YB = 2(7,
Y(T= 0*  1) F,
Yff= ,iA\
YA'= 0.
Conversely if these equations are satisfied the function will be a covariant.
I assume that il is a function of the degree and of the weight ^ (md — /i), satisfying
the condition
141] A SECX)ND MEMOm UPON QUANTICS. 255
and I represent by YA, FM, Y^A, &c. the results obtained by successive operations
with Y upon the function A, The function Y'A will be of the degree and of the
weight ^(mO  fi) + 8, And it is clear that in the series of terms YA, FM, FM, &e.,
we must at last come to a term which is equal to zero. In fact, since m is the
greatest weight of any coeflBcient, the weight of F* is at most equal to m0, and therefore
if i(md — /i)+«>m5, or 8 > ^(mO ^^ fi), we must have Y'A=0.
Now writing for greater simplicity XY instead o{ X,Y, and so in similar cases, we
have, as regards Y'A,
XYYX = fi28.
Hence
^od consequently
i*5iinilarly
^ therefore
d again,
therefore
^nerally
{XY'YX)A = fiA,
XYA=^YXA'\lj^A=ij^A,
(ZF FX) F^ = (/i  2) Fil,
X Y^A = FZ Fil + (/i  2 ) Fil
= /aF4 + (a* 2) F.4 = 2(/i l)Fil.
(XFFZ)F«il=(/A4)FM,
XY^A = YXY^A + (/i  4) Y^A
= 2(/t 1) F«il + 0i 4)FM = 3 (/i 2) FM,
ZF'^=«(/A« + l)FM.
^ce putting « = ^+l, /i + 2, &c., we have
ZF'*+M = 0,
XYi'^^A =  (/i + 2) 1 . F'*+'^,
ZF'*+M=(/i + 3)2.F'*«J,
&c,
F^+M = ;
y^*^ unless this be so, La if F''+*4 + 0, then from the second equation Z F'^+M + 0, and
^^^fore Yf'^A^O, from the third equation ZF'*+»=NO, and therefore Z'^+M^O, and so
^ dd infinitum^ Le. we must have Y'^'^^A = 0.
^^^^^ations which show that
256 A SECOND MEMOIR UPON QUANTIC8. [141
81. The Buppoeitiona which have been made as to the function A, give therefore
the equations
XA =0.
XYA =,iA.
XY*A = 2(ji\)YA,
XYi^A'=(tYi'*'A,
Yi'+'A = ;
and if we now assume
the sratem becomes
XA'=0,
XB = fiA,
XC = (jil)B.
J A' = F,
YA' = 0;
80 that the entire system of equations which express that (A, B.,.B, ^^^^f vY ^
a covariant is satisfied ; hence
Theorem. Given a quantic (a, 6, ...6\ a'$a?, y)***; if il be a function of the
coefficients of the degree and of the weight ^ (mO — fi) satisfying the condition
XA =0, and if B, C, ... jB", 4' are determined by the equations
B^YA, C = ^YB,,..A' = ^YB\
then will
(A, B,...B, A\x, yY
be a covariant.
In particular, a function A of the degree and of the weight ^m^, satisfying the
condition XA = 0, will (also satisfy the equation YA = and vdll) be an invariant.
32. I take now for A the most general function of the coefficients, of the degree 6
and of the weight \ {mO — /x) ; then XA is a function of the degree 6 and of the weight
^(m^ — /i)— 1, and the arbitrary coefficients in the function A are to be determined
so that XA =■ 0. The number of arbitrary coefficients is equal to the number of
terms in A^ and the number of the equations to be satisfied is equal to the number of
terms in XA ; hence the number of the arbitrary coefficients which remains indeter
minate is equal to the number of terms in A less the number of terms in XA ; and
since the covariant is completely determined when the leading coefficient is known,
141]
A SECOND MEMOIR UPON QUANTICS.
257
the diflference in question is equal to the number of the asyzygetic covariants, i.e. the
number of the asyzygetic covariants of the order fi and the degree is equal to the
number of terms of the degree and weight \ (mO — fi), less the number of terms of
the degree d and weight J(md— /a)— 1.
83. I shall now give some instances of the calculation of covariants by the method
just explained. It is very convenient for this purpose to commence by forming the
literal parts by Arbogast's Method of Derivations : we thus form tables such as the
following :—
a
b
c
a*
ah
ac
6«
be
6»
a
b
c
' !
ah
6«
ad
be
bd
ed
d" !
rt*
a^b
a*c
aH
abd
aed
cuP
bd"
ed"
(P
ah^
ahe
b"
ac"
b^e
b^d
be"
bed
c'd
a*
M
a»<J
aH
a^bd
a^ed
a^d"
abcP
a^<P
ad^
bd^
ccP
d*
a«6»
a^be
a^c"
ab'd
abed
ac'd
b^d"
bcd^
c^rf*
ab*
ah^c
6*
abc"
b^c
bH
6V
b^ed
be"
bc'd
eH
a
b
c
d
e
a»
ab
ac
ad
M
be
bd
ed
d'
V
he
bd
ed
<?
C. II.
33
258
A SECOND MEMOIB UPON QUANTICS.
[141
a'
a*6
a^c
a^d
a««
abe
ace
ode
oe*
W
ce»
de*
^
ah*
abc
abd
acd
cuP
bee
bde
cde
d^.
&»
b*d
be"
bh
bed
c'd
d^

<?
34. Thus in the case of a cubic (a, 6, c, d$a?, y)*, the tables show that there will
be a single invariant of the degree 4. Represent this by
•\' Babcd
4 Cad"
+ Dl^d
which is to be operated upon with a36 + 263c + 3c9d This gives
Le.
E^
+ B
+ 6i4
a^cd
+ 3/>
+ 25
ab\l
+ '2E
+ 6C
+ 45
putting .
+ 35
+ 35
abc*
b'c
5 = 0, &c.;
or
zl
= 1, we
fi
nd 5
= (], C=4, Z> = 4.
— 3, and the invariant is
— 6a6cd
+ 4ac»
+ 46»d
36V.
Again, there is a covariant of the order 3 and the degree 3. The coefficient of .r*
leading coefficient is
Aa*d
•i Babe
which operated upon with a3e> + 2ft3c4 3c9d, gives
+ 5
+ 3C
1
+ 2Z?
+ 3i4
a'e
ab'
141]
A SECOND MEMOIR UPON QUANTICS.
259
i.e. B^SA =0, 3C+aB = 0; or putting ^ = 1, we have 5 = 3, C = 2, and the leading
coefficient is
 Sabc
+ 2 6».
The coefficient of ah/ is found by operating upon this with (369a + 2cdi, + ddc), this
gives
abd
ac'
b'c
+ 6
9
6
+ 12
3
i.e. the required coefficient of a^y is
Sabd
 6ac»
43 6»c;
and by operating upon this with J (369a + ^cdb + dde\ we have for the coefficient of a?y^
acd
b'd
be'
4. »
+ If
9
+ 3
+ 6
6
4. »
i.e. the coefficient of xi/* is
— 3 acd
+ 66»d
~3 6c».
Finally, operating upon this with ^ (369a + 2c9ft + d9c)> we have for the coefficient of y*,
1
3
+ 8
2
2
i.e. the coefficient of y* is
and the covariant is
+ 36cd
2c»,
ad'
bed
( abe3
V +2
oc* 6
6*c +3
h*d +6
ftc" 3
1
6e(^ + 3
c» 2
$a^» yf
[I now write the numerical coefficients after instead of before the literal terms.]
33—2
260
A SECOND MEMOIR UPON QU ANTICS.
[141
I have worked out the example in detail as a specimen of the most convenient method
for the actual calculation of more complicated covaiiants^
35. The number of terms of the degree and of the weight q is obviously
equal to the number of wajrs in which q can be made up as a sum of 6 terms
with the elements (0, 1, 2, ...m), a number which is equal to the coeflBcient of aflg^ in
the development of
___^ 1
and the number of the asyzygetic covariants of any particular degree for the quantic
(♦$^> y)** can therefore be determined by means of this development. In the case of
a cubic, for example, the function to be developed is
{\ " z){l  xz) {\ a^z) {I a?zy
which is equal to
where the coeflScients are given by the following table ; on account of the symmetry,
the series of coefficients for each power of ^r is continued only to the middle tenn or
middle of the series.
1
1
1
1
1
2
2
1
1
2
3
3
1
1
2
3
4
4
5
1
I
2
3
4
5
6
6
1
1
2
3
4
5
7
7
8
8
(0)
(1)
(2)
(3)
(4)
(5)
(6)
^ Note added Feb. 7, 1856.— The following method for the calculation of an invariant or of the leading
coefficient of a covariant, is easily seen to be identical in principle with that given in the text Write down
all the terms of the weight next inferior to that of the invariant or leading coefficient, and operate on each
of theee separately with the symbol
ind. 6 .  + 2 ind. c . r +...(»» 1) ind. 6* . ^ ,
where we are first to multiply by the fraction, rejecting negative powers, and then by the index of the proper
letter in the term so obtained. Equating the results to zero, we obtain equations between the terms of the
invariant or leading coefficient, and replacing in these equations each term by its numerical coefficient in the
141]
A SECOND MEMOIR UPON QUANTICS.
261
and from this, by subtracting from each coefficient the coefficient which immediately
precedes it, we form the table:
(0)
L
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
(
}
1
1
1
1
1
2
1
(2)
(3)
(5)
The successive lines fix the number and character of the covariants of the degrees
0, 1, 2, 3, &C. The line (0), if this were to be interpreted, would show that there is a
single covariant of the degree ; this covariant is of course merely the absolute con
stant unity, and may be excluded. The line (1) shows that there is a single covariant
of the degree 1, viz. a covariant of the order 3; this is the cubic itself, which I
represent by U. The line (2) shows that there are two asyzygetic covariants of the
degree 2, viz. one of the order 6, this is merely JJ\ and one of the order 2, this I
represent by H, The line (3) shows that there are three asyzygetic covariants of the
degree 3, viz. one of the order 9, this is JJ* ; one of the order 5, this is UH, and one of
the order 3, this I represent by 4). The line (4) shows that there are five asyzygetic
covariants of the degree 4, viz. one of the order 12, this is U^\ one of the order 8,
this is V^H\ one of the order 6, this is JT*; and one of the order 0, ie. an invariant,
this I represent by V. The line (5) shows that there are six asyzygetic covariants of
the degree 5, viz. one of the order 15, this is IT" ; one of the order 11, this is U^H ;
one of the order 9, this is I7*4> ; one of the order 7, this is UH*\ one of the order 5,
this is H^ ; and one of the order 3, this is V CT. The line (6) shows that there are 8
asyzygetic covariants of the degree 6, viz. one of the order 18, this is U^\ one of the
inyariant or leading coefficient, we have the equations of connexion of these nomerioal coefficients. Thns, for
the dlBcriminant of a cnhic, the terms of the next inferior weight are ahd^ ab^d, abc\ &*e, and operating on
each of these separately with the symhol
ind. 6 .  + 2 ind. c . ■=• + 3 ind. d .  ,
a b c
we find
abed
+ 6a«d'
3bH
+ 2abed
2 6»c«
+ 6ac«
■\Sabcd
+ 4fc»c>
+ 8 6»d
and equating the horizontal lines to zero, and assuming a'd':=l, we have a^=l, abed= 6, ac^=4, b*d=if
6M=  3, or the value of the discriminant is that given in the text.
262
A SECX)ND MEMOIR UPON QU AN TICS.
[141
order 14, this is U*H \ one of the order 12, this is 17** ; one of the order 10, thi^ is
U^H^\ one of the order 8, this is UH^\ two of the order 6 (Le. the three covariants
ff', ^ and Vf/"* are not asyzygetic, but are connected by a single linear equation or
syzygy), and one of the order 2, this is VH. We are thus led to the irreducible
covariants U, JT, ^, V connected by a linear equation or syzygy between H*, ^' and
VfT', and this is in fact the complete system of irreducible covariants; V is therefore
the only invariant.
36. The asyzygetic covariants are of the form U^H^V^^ or else of the form
U^H^V^^; and since U, H, V are of the degrees 1, 2, 4 respectively, and ^ is of the
degree 3, the number of asyzygetic covariants of the degree m of the first form is
equal to the coefficient of a^ in 1 r (1 — x) (1 — a^) (1 — a^), and the number of the
asyzygetic covariants of the degree m of the second form is equal to the coefficient
of w'^ in a^^(l— a:) (1 — ic")(l — a?*). Hence the total number of asyzygetic covariants is
equal to the coefficient of »"* in (1 + a;*) ^ (1 — x) (1 — x*) (I — a^), or what is the same
thing, in
and conversely, if this expression for the number of the asyzygetic covariants of the
degree m were established independently, it would follow that the irreducible invariants
were four in number, and of the degrees 1, 2, 3, 4 respectively, but connected by
an equation of the degree 6. As regards the invariants, eveiy invariant is of the
form V^, i.e. the number of asyzygetic invariants of the degree m is equal to the
coefficient of ic*" in , ^ , and conversely, fiom this expression it would follow that
there was a single irreducible invariant of the degree 4.
37. In the case of a quartic, the function to be developed is:
1
(1  £r) (1  arz) (1  a:»z) (1  a^^) (1  a:*^) '
and the coefficients are given by the table.
(0)
0)
(2)
(3)
(4)
(5)
(6)
1
1
1
1
1
1
2
2
3
1
1
2
3
4
4
5
1
1
2
3
5
5
7
7
8
1
1
2
3
5
6
8
9
11
11
12
1
1
2
3
5
6
9
10
13
14
16
16
18
141]
A SECOND MEMOIR UPON QUANTICS.
263
and subtracting from each coefficient the coefficient immediately preceding it, we have
the table :
1
(0)
1
(1)
1
1
1
(2)
1
1 1
1
1
1
1
(3)
1
1
t
1
2
2
(4)
1
1
1
1
2
3
1
1
2
1
2
1
(5)
1
1
1
2
3
1
2
2
(6)
the examination of which will show that we have for the quartic the following
irreducible covariants, viz. the quartic itself U; an invariant of the degree 2, which I
represent by / ; a covariant of the order 4 and of the degree 2, which I represent by H ;
an invariant of the degree 3, which I represent by J; and a covariant of the order 6
and the degree 3, which I represent by <I> ; but that the irreducible covariants are
connected by an equation of the degree 6, viz. there is a linear equation or syzygy
between <!>», PH^, PJH^U, IJ^HU^ and J^U^\ this is in fact the complete system of
the irreducible covariants of the quartic: the only irreducible invariants are the
invariants /, J.
38. The asyzygetic covariants are of the form U^I'^H^J*, or eke of the form
UpI'^H^'J*^, and the number of the asyzygetic covariants of the degree m is equal to
the coefficient of a^ in (1 + ir')^(l — a:)(l — a:^)*(l — a^), or what is the same thing, in
1A^
(ij')(i'a^y(ix'y'
and the asyzygetic invariants are of the form /pJ^, and the number of the asyzygetic
invariants of the degree m is equal to the coefficient of af^ in 1t(1 — a;*)(l — a^).
Conversely, if these formulae were established, the preceding results as to the form
of the system of the irreducible covariants or of the irreducible invariants, would
follow.
39. In the cas^ of a quintic, the function to be developed is
(I " z)(l  xz){l  a^z)(l ^ .z^z) (I a^z)(la^z)'
and the coefficients are given by the table :
264
A SECOND MEMOIR UPON QUANTICS.
[1
1
1
1
1
1
1
2
2
3
3
1
1
2
3
4
5
6
6
1
1
2
3 5
6
8
9
11
11
12
1
1
2
3
5
7
9
11
U
16
18
19
20
(0)
(1)
(2)
(3)
(4)
(5)
and subtracting from each coefficient the one which immediately precedes it, we h
the table :
1
1
1
1
1
1
1
1
1
1
1
1
(0)
0)
(2)
(3)
(4)
(5)
We thus obtain the following irreducible covariants, viz. :
Of the degree 1 ; a single covariant of the order 5, this is the quintic itself
Of the degree 2 ; two covariants, viz. one of the order 6, and one of the order 2
Of the degree 3 ; three covariants, viz. one of the order 9, one of the order 5, i
one of the order 3.
Of the degree 4 ; three covariants, viz. one of the order 6, one of the order 4, i
one of the order (an invariant).
Of the degree 6 ; three covariants, viz. one of the order 7, one of the order 8, i
one of the order 1 (a linear covariant).
These covariants are connected by a single syzygy of the degree 5 and of
order 11 ; in fact, the table shows that there are only two asyzygetic covariants
this degree and order; but we may, with the abovementioned irreducible covaria
of the degrees, 1, 2, 3 and 4, form three covariants of the degree 5 and the or
11 ; there is therefore a syzygy of this degree and order.
141] A SECOND MEMOIR UPON QU ANTICS. 265
40. I represent the number of ways in which q can be made up as a sum of
m terms with the elements 0, 1, 2, . . . m, each element being repeatable an indefinite
number of times by the notation
P(0, 1, 2, ...m/j,
and I write for shortness
P'(0, 1, 2, ...m/? = P(0, 1, 2...m/gP(0, 1, 2 ... m/(j 1).
Then for a quantic of the order m, the number of asyzygetic covariants of the degree
and of the order /i is
P'(0, 1, 2...m/i(m5A*).
In particular, the number of asyzygetic invariants of the degree 6 is
P'(0, 1, 2...mf^e.
To find the total number of the asyzygetic covariants of the degree 6, suppose
first that md is even ; then, giving to /x the successive values 0, 2, 4, . . . mO, the
required number is
P{^e) PQm^l)
+ P(im5l)~P(Jm^2)
+ P(2) P(l)
+ P(1)
= P(W)>
L e. when md is even, the number of the asyzygetic covariants of the degree 6 is
P(0, 1, 2...m/imd;
and similarly, when md is odd, the number of the asyzygetic covariants of the degree
d is
P(0, 1, 2, ...m)•i(»^5l)•
But the two formulae may be united into a single formula; for when inB is odd \md
is a fraction, and therefore P{^ff) vanishes, and so when mB is even \{m6—\) is a
fraction, and Pi(m5 — 1) vanishes; we have thus the theorem, that for a quantic of
the order m\
The number of the asyzygetic covariants of the degree 6 is
P(0, 1, 2...m)«im5 + P(0, 1, 2, ... m)«Kwid !)•
41. The functions P{\m0\ &c. may, by the method explained in my "Researches
on the Partition of Numbers," [140], be determined as the coefficients of a^ in certain
functions of a? ; I have calculated the following particular cases : —
Putting, for shortness,
P'(0, 1, 2,... m)* ^m5 = coefficient of in <f>m,
C. II. 34
266 A SECOND MEMOIR UPON QUANTICS. [141
then 4>2 =
then
^ =
1
^ (la;»)(l~aj»)'
Afi  (1 a? )(l+a?a:'a?* je'+jg^ + a:»)
'''^ "■ (l««)«(la:»)(la;*)(la:») '
.Q^ (la?)(l+a?d^~a;* + a^ + af + g» + g» + a^<>g» + a;^» + a:^8)
(la;»)«(la:»)«(l~j;*)(la:»)(la;')
P(0, 1, 2, ... m)* ^md = coefficient of a:^ in ^m,
then i/^2 =
Vr3 =
l+ar*
(l~a;»)'(la;*)'
''' (la?)«(la;»)(l~a^)*
1 + a:* + 6a:* + 9^* + 12g» + Qx"^ k 6a^ + x"^ ^ a^""
P(0, 1, 2, ... m)*^md— 1) = coefficient of a:;* in ^m,
''^ (la»)«(la;*)'
''^ (la:»)»(l«*)(l «•)(! ic')
And from what has preceded, it appectrs that for a quantic of the order m, the
number of asyzygetic covariants of the degree ^ is for m even, coefficient a^ in ^i,
and for m odd, coefficient a^ in (yp^m + yfr^m); and that the number of asyzygetic
invariants of the degree is coefficient a:^ in ifym. Attending first to the invariants:
42. For a quadric, the number of asyzygetic invariants of the degree is
1
coefficient of in
la;»'
which leads to the conclusion that there is a single irreducible invariant of the
dijgree 2.
141] A SECOND MEMOIR UPON QUANTICS. 267
43. For a cubic, the number of asyzygetic invariants of the degree is
coefficient a? in .j — .,
1 — ar
i.e. there is a single irreducible invariant of the degree 4.
44. For a quartic, the number of asyzygetic invariants of the degree is
coefficient of in (i _ J^^^ _ ^) .
i.e. there are two irreducible invariants of the degrees 2 and 3 respectively.
45. For a quintic, the number of asjrzygetic invariants of the degree is
l^af + a^
coefficient «• in
(la;*)(lir«)(la^)'
The numerator is the irreducible factor of 1 — ic", Le. it is equal to (1 — a;") (1 — «•)
r(l — a:") (1 — a:") ; and substituting this value, the number becomes
1 — ^
coefficient a:^ in
(la;*)(la*)(la:")(ld5»)'
Le. there are in all four irreducible invariants, which are of the degrees 4, 8, 12 and
18 respectively; but these are connected by an equation of the degree 36, i.e. the
square of the invariant of the degree 18 is a rational and integral function of the
other three invariants; a result, the discovery of which is due to M. Hermite.
46. For a sextic, the number of asyzygetic invariants of the degree is
/»• X a  (l'x)(l'\x — a^ — a^ — a:^ + w' + a^)
coefficient ar in ^^ — /■. ^x, /■. ijwi zitti =;\ •
(1  a:;*)* (1  a^) (1  a?*) (1  of)
the second factor of the numerator is the irreducible fiswtor 1 — a:**, i. e. it is equal
to (laJ»)(la:»)(la^)(l~a:»)4(la:")(la^<>)(laj»)(la;); and substituting this
value, the number becomes
la:»
coefficient as* in
(1  a:') (1  a?*) (1  a;*) (1  x"') (1  ai'') '
Le. there are in all five irreducible invariants, which are of the degrees 2, 4, 6, 10
and 15 respectively; but these are connected by an equation of the degree 30, i.e.
the square of the invariant of the degree 15 is a rational and integral function of
the other four invarianta
47. For a septimic, the number of asyzygetic invariants of the degree is
la:«f2a;»a^* + 6a^ + 2a^* + 6a:^« + 2a:" + 5a:»a;«+2a;^a:« + a*
coefficient a;* in
(1  a?*)(l ~a^)(l a;«)(l a;^»)(l  x"')
34—2
268 A SECOND MEMOIR UPON QUANTICS. [141
the numerator is equal to
(1  a:)(l a:»)*(l ir^»)(l a:")*(l «")^....
where the series of factors does not terminate; hence [incorrect, see p. 253] the number
of irreducible invariants is infinite; substituting the preceding value, the number of
asyzygetic invariants of the degree is
coefficient a:^ in (1  aj*)> (1  a^y' (1 " ^^ (1  ^^ V • •
The first four indices give the number of irreducible invariants of the corresponding
degrees, i.e. there are 1, 3, 6 and 4 irreducible invariants of the degrees 4, 8, 12 and
14 respectively, but there is no reason to believe that the same thing holds with
respect to the indices of the subsequent terms. To verify this it is to be remarked,
that there are 1, 4, 10 and 4 asyzygetic invariants of the degrees in question respect
ively; there is therefore one irreducible invariant of the degree 4; calling this Z4,
there is only one composite invariant of the degree 8, viz. X/; there are therefore
three irreducible invariants of this degree, say Xg, Xg', Xg". The composite invariants
of the degree 12 are four in number, viz. X^*, X^X^, X^X^\ X^X^\ and these cannot he
connected by any syzygy, for if they were so, X^\ Xg, Xg', X," would be connected by a
syzygy, or there would be less than 3 irreducible invariants of the degree 8. Hence
there are precisely 6 irreducible invariants of the degree 12. And since the irreducible
invariants of the degrees 4, 8 and 12 do not give rise to any composite invariant of
the degree 14, there are precisely 4 irreducible invariants of the degree 14
48. For an octavic, the number of the asyzygetic invariants of the degree is
coefficient ar* in
(1 a;)(l f xa^ a^ + a^ + a?' h a;* + ic* + a?**  a?*'  «*' h «" + «*•) .
{lx'yil a:»)«(l a:*)(l ar»)(l ^af)
and the second factor of the numerator is
(1 a:)»(l a:»)(l ar')»(l a:V(l ^)"' (laf)'(l ar»V(l a;**)(l a?*0(l ^') 
where the series of factors does not terminate, hence [incorrect] the number of irreducible
invariants is infinite. Substituting the preceding value, the number of the asyzygetic
invariants of the degree is
roeffla;* in (1  j;*)' (1  ar»)i (1  ar*)^!  4^)> (1  a:«)i (1  ar^^
There is certainly one, and only one irreducible invariant for each of the degrees
2, 3, 4, 5 and 6 respectively; but the formula does not show the number of the irre
ducible invariants of the degrees 7, &c. ; in fact, representing the irreducible inva
riants of the degrees 2, 3, 4, 5 and 6 by X„ X,, X4, X5, Xg, these give rise to 3 com
posite invariants of the degree 7, viz. X,X^j, XjXj, X,X4, which may or may not be
connected by a syzygy; if they are not connected by a syzygy, there will be a single
irreducible invariant of the degree 7 ; but if they are connected by a sjzygy, there
will be two irreducible invariants of the degree 7 ; it is useless at present to pursue
the discussion further.
\
141] A SECOND MEMOIR UPON QUANTICS. 269
Considering next the covariants, —
49. For a quadric, the number of asyzygetic covariants of the degree is
1
coeflScient of in
(\x){\<d^y
i.e. there are two irreducible covariants of the degrees 1 and 2 respectively; these
are of course the quadric itself and the invariant.
50. For a cubic, the number of the asyzygetic covariants of the degree 6 is
coefficient of m ^^_Jy^^^^) •
The first £Etctor of the numerator is the irreducible factor of
la^, = (la^)(lfl?),
snd the second factor of the numerator is the irreducible factor of
laj*, = (laj*)^(la^);
^substituting these values, the number is
coefficient of in
(1 a?)(l a;»)(l a:»)(l iT*)'
i.e. there are 4 irreducible covariants of the degrees 1, 2, 3, 4 respectively; but these
^re connected by an equation of the degree 6; the covariant of the degree 1 is the
oubic itself U, the other covariants are the covariants already spoken of and repre
sented by the letters JJ, 4> and V respectively {H is of the degree 2 and the order 3,
<X> of the degree 3 and the order 3, and V is of the degree 4 and the order 0,
x.e. it is an invariant).
51. For a quartic, the number of the asyzygetic covariants of the degree 6 is
coefficient sfi in
(l«)«(laj»)(laj»)*
^he numerator of which is the irreducible factor of l—af^ i.a it is equal to
(1 — «•) (1 — «) r (1 — oj*) (1 — aj*). Making this substitution, the number is
1 — ir*
coefficient of in
(la?)(la^)»(la^)«'
X.e. there are five irreducible covariants, one of the degree 1, two of the degree 2,
^,nd two of the degree 3, but these are connected by an equation of the degree 6.
*irhe irreducible covariant of the degree 1 is of course the quartic itself U, the other
irreducible covariants are those already spoken of and represented by /, H, J, 4>
^"espectively (/ is of the degree 2 and the order 0, and J is of the degree 3 and
^he order 0, Le. / and J are invariants, H is of the degree 2 and the order 4, 4>
xs of the degree 3 and the order 6).
270 A SECOND MEMOIR UPON QUANTICS. [141
52. For a quintic, the number of irreducible covariants of the degree is
the numerator of which is
(l+a?)»(la?H2a;»Hir«H2a^HS«» + ««H5a;'Ha^H8a;* + 2a^'Ha?" + 2a;"a^
the first £Eu;tor is (1 — ai)~^ (1 — a^y, the second factor is
(1 ^)(1 a^)«(l aj»)»(l a:*)^(l af)'^(l afiy{lx'y(l''a*y(l^a^y(lx'^)^(l^a!''y^...,
which does not terminate ; hence [incorrect] the number of irreducible oovariants is
infinite. Substituting the preceding values, the expression for the number of the
asyzygetic covariants of the degree 6 is
coeff. ^in (lx)^ (1a'«)«(1 a;»)«(l .d?«)»(l a?*)«(l «•)*(!  x^)^ (I  a*f (I j^)i(l ar»«)«(l ^»)"**....
which agrees with a previous result: the numbers of irreducible covariants for the
degrees 1, 2, 3, 4 are 1, 2, 3 and 3 respectively, and for the degree 5, the number
of irreducible covariants is three, but there is one syzygy between the composite
covariants of the degree in question ; the difference 3 — 1 » 2 is the index taken with
its sign reversed of the factor (1 — aj")~*.
53. I consider a system of the asyzygetic covariants of any particular degree and
order of a given quantic, the system may of course be replaced by a system the terms
of which are any linear functions of those of the original system, and it is necessary
to inquire what covariants ought to be selected as most proper to represent the
system of asyzygetic covariants; the following considerations seem to me to furnish
a convenient rule of selection. Let the literal parts of the terms which enter into
the coeflScients of the highest power of x or leading coefficients be represented by
Ma, Mp, My,,., these quantities being arranged in the natural or alphabetical order;
the first in order of these quantities M, which enters into the leading coefficient of a
particular covariant, may for shortness be called the leading term of such covariant,
and a covariant the leading term of which is posterior in order to the leading terra
of another covariant, may be said to have a lower leading term.
It is clear, that by properly determining the multipliers of the linear functions we
may form a covariant the leading term of which is lower than the leading term of
any other covariant (the definition implies that there is but one such covariant); call
this 0. We may in like manner form^ a covariant such that its leading term is lower
than the leading term of every other covariant except 0i ; or rather we may form a
system of such covariants, since if 4>, be a covariant having the property in question,
<l)j + Ar0i will have the same property, but k may be determined so that the covariant
shall not contain the leading term of 6i, i.e. we may form a covariant B, such that
its leading term is lower than the leading term of every other covariant excepting
01, and that the leading term of ©i does not enter into 0,; and there is but one such
covariant, 0,. Again, we may form a covariant 0, such that its leading term is lower
than the leading term of every other covariant excepting 0i and 0,, and that the
141]
A SECX)ND MEMOIR UPON QUANTIC8.
271
leading terms of Bi and B, do not either of them enter into B, ; and there is but
one such co variant, B,. And so on, until we arrive at a covariant the leading term
of which is higher than the leading terms of the other covariants, and which does
not contain the leading terms of the other covariants. We have thus a series of
covariants Si, B„ B,, &c. containing the proper number of terms, and which covariants
may be taken to represent the asyzygetic covariants of the degree and order in question.
In order to render the covariants B definite as well numerically as in regard to
sign, we may suppose that the covariant is in its least terms (Le. we may reject
numerical £Etctors common to all the terms), and we may make the leading term
positive. The leading term with the proper numerical coeflScient (if different from
unity) and with the proper power of x (or the order of the function) annexed, will,
when the covariants of a quantic are tabulated, be sufficient to indicate, without any
ambiguity whatever, the particular covariant referred to. I subjoin a table of the
covariants of a quadric, a cubic and a quartic, and of the covariants of the degrees
1, 2, 3, 4 and 5 respectively of a quintic, and also two other invariants of a quintic.
[Except for the quantic itself, the algebraical sum of the numerical coefficients
in any column is =0, viz. the sum of the coefficients with the sign + is equal to
that of the coefficients with the sign — , and I have as a numerical verification
inserted at the foot of each column this sum with the sign ±].
(
Covariant Tables (Noa 1 to 26).
No. 1. No. 2.
5 ^» yY'
a+l
6 + 2
c + 1
=fc 1
The tables Noa 1 and 2 are the covariants of a binary qi^adric. No. 1 is the
quadric itself; No. 2 is the quadrin variant, which is also the discriminant.
No. 3. No. 4.
H^. UY
\^y i/Y
± 1
± 1
± 1
No. o.
No. G.
ac/f 1
a6rf+3
acrf3
OiP 1
a6c~ 3
ac" 6
b*d +6
bed + 3
A» +2
b^c +3
bc^ 3
c» 2
5, yr
:L H
:i^6
± 6
± 3
a'd'
+ 1
abed
6
a^
+ 4
bH
+ 4
6»c«
 3
± y
The tables Noa 3, 4, 5 and 6 are the covariants of a binary cubic. No. 3 is the
cubic itself; No. 4 is the quadricovariant, or Hessian; No. 5 is the cubicovariant ;
No. 6 is the invariant, or discriminant. And if we write No. 3 = IT, No. 4 = if.
No. 5 = *, No. 6 = V,
272
A SECOND MEMOIR UPON QUANTICS.
then identically,
No. 7.
a+1
6 + 4
C4 6
(^ + 4
« + l
$«.y)*
No.
8.
00 +
1
W
4
c« +
3
No. 9.
(
±*
oe + l
6«^1
ad + 2
6c2
06+ 1
6rf+2
c«3
6« +2
crf2
6J+1
c« 1
$^y)*
Jr 1
±2
±3
db2
± 1
No.
10.
ac6
+ 1
OflP
1
b^e
1
bed
+ 2
c»
 1
No. 11.
±s
o«rf+l
o*« + 1
o56 + 5
oe«
a(f0< 5
a^ 1
6tf«  1
o6c3
o6i+2
aed^ 15
o^lO
6c0 + 15
6<fe.2
C6fe f 3
6» +2
oc«9
6y + 10
6»tf + 10
bd'lO
c"tf + 9
rf» 2
6»c +6
6c*
bed
er • . •
<?d
crf»6
^
^8
:1:9
:k 15
dblO
:l:16
:t9
±8
No.
12.
oV
+
1
o«6c£e»
—
12
oVe*
—
18
a^cd^e
f
54
a^d^
—
27
ab'e^
+
54
ab'd'e
—
6
abc^de
—
180
abecP
+
108
oc^
+
81
aA?d}
— i
54
6V
—
27
6'o<^
+
108
6»ci»
—
64
6V«
—
54
6V(i*
+
36
6c*rf
• • •
c«
• • •
^442
The tables Nos. 7, 8, 9, 10 and 11 are the irreducible covariants of a qu
No. 7 is the quartic itself; No. 8 is the quadrinvariant ; No. 9 is the quadricova]
or Hessian; No. 10 is the cubinvariant ; and No. 11 is the cubicovariant. The
No. 12 is the discriminant. And if we write No. 7 = CT, No. 8 = /, No. 9
No. 10 = J, No. 11 = *, No. 12 = V,
then the irreducible covariants are connected by the identical equation
jir»/f7«fr+4fr» + 4)« = o,
and we have
V = /»27J«.
141]
A SECOND MEMOIR UPON QUANTICS.
273
[The Tables Nos. 13 to 24 which follow, and also Nos. 25 and 26 which are given
in 143 relate to the binary quintic. I have inserted in the headings the capital letters
A, B, . . . L and also Q and Q' by which I refer to these covaiiants of the quintic. A is
the quintic itself, C is the Hessian, G is the quartin variant, J a linear covariant: Q is
the simplest octinvariant, and Q' is the discriminant. As noticed in the original memoir
we have AI + BFCE = 0; and Q' = G«  128 Q, only the coefficient 128 was by mistake
given as 1152.]
A. No. 13.
(
a + 1
6 + 5
c+10
rf+10
6 + 5
/+i
Xa^y)'
B. No. 14.
(
ae+1
bdi
c* +3
a/+l
6e3
ed+ 2
kf+1
ce4:
d»+3
tn^ y)*
:1:4 :1:8 ±4
C. No. 15.
(
oc + 1
6* 1
ad+3
be 3
ae + 3
W+3
c" 6
(l/'+l
6« +7
crf8
6/ + 3
cd +3
cP 6
C/ + 3
deZ
d/^l
e» 1
H^ yY
db 1
=k8
:t 6 ^S ±6
D. No. 16.
^B
=fc 1
(
a^ ...
06/"+ 1
a4f*\
o«^ ...
ace + 1
ode— 1
w? 1
6<ir+i
atP1
jyi
6c^l
W 1
Va 1
&C0 + 1
5<2«il
«yi
hcd+2
M»+l
«»« +1
ed0 + 2
c* 1
c'rfl
aPl
<i» 1
$«, y)'
:^8
8
8
8
K
No. 17.
ay+1
ay+ 5
acf^ 2
adf 2
cw/ 5
qr 1
a6d  5
ace  16
ocie12
oa*  8
64/'+ 16
6i2/+5
, 0(^+2
V 6*^ + 8
a<i'+ 6
6!/'+ 8
6c/ +12
6«« + 9
c(i/'2
6»6  9
6c« 38
6(20+38
c*/ 6
C6* 8
he 6
5c(2+38
M> + 72
i?e 72
ce2038
(^6 +6
c» 24
c'rf32
c«P + 32
d» +24
:1:11
:t49
:1:S2
:1>S2
:i:49
dbii
$^. yy
F. No. 18.
I
(i»rf+l
a6c3
6» +2
aV +
2
a6<;+ 1
oa" 12
6»c + 9
a^f^ 1
a6a + ll
occf 34
6»rf+16
6c» + 6
a6/+ 7
ace — 8
arf«34
6*6 +29
bed 2
c* + 8
a^+ 5
acfo40
6*/ + 16
6c6 + 47
6c^44
ed + 16
o^/ 5
ac« 16
6c/ + 40
bde^7
&e +44
c(P 16
oc/ 7
6(f/+ 8
6«» 29
c«/ + 34
e£f0+ 2
d»  8
<!/•• 1
6e/ll
«i^+34
(»» 16
cPtf  6
6/« 2
<»/ 1
rf»/+12
(fo«  9
C/^1
cis/'+S
e» 2
:l:8 :& 12
C. II.
:k84
:1:44
:k84
84
^44
:S=84
^\2
35
$«^y)'
274
A SECOND MEMOm UPON QUANTICS.
[141
G. No. 19.
«!/•*+ 1
a6(?/ 10
acdf^ 4
occ* + 16
ad^e 12
hHf^ 16
6V + 9
6cy 12
hcde  76
hd" +48
c^tf + 48
c^d^  32
H. No. 20.
^ 142
a«4/'+ 1
aV + 2
ay « + 1
a6/«+ 3
a^«+ 1
a V
a6^ 4
a6«/  4
acef 4
adef 3
aftc/ 3
aM 10
occj/' 2
aciy 2
a«« + 2
aftc2s  5
ac»/ 2
ace" + 4
oc^ + 4
6y« ...
ao*« + 10
oecfe + 24
a^i'e
6V  10
beef  5
/ acd^ — 4
0(^12
ft«rf/+ 4
6ai^+ 24
bd"/^ 10
^ 6'/ + 2
6»</ + 4
6V  9
ftc«» + 16
M«»  5
h^ce  5
W« +16
W/ ...
6c^e 22
c»4/ 4
6«<^ +14
ftc«e 22
6ccfo + 50
cy 12
c*fi" + 14
he'd  16
6cc^  4
W 36
c«<fo  4
crf««  16
c* + 6
i
c«rf + 8
c»« 36
cW +28
cc^ + 8
rf* + 6
dbSS
db54
j=87
db54
=k83
5*. i^)*
I. No. 21.
ttV* + 1
a«ei/+ 2
a*^
ay« ...
abp ..,
ac/« 2
«(/•• 1
a*(ie  1
aV  2
abd/ + 2
cU>e/
ac«/2
culef
a^y +1
' a6yl
a^/10
a6e" 2
acdf ...
acV+l
ae> + 2
her + 1
abce  2
a6c^+ 10
ocyl
a<j«» 20
arf(5» + 1
6y» + 2
6cfe/+2
a6«/* + 4
ac**
acde2
arf*« + 20
6y +2
beef
bSf^ 10
6«* 3
. ac*rf  1
acd^
cuP +S
b^d/ + 20
M^+2
c V  4
< 6»e +3
6y  2
b'cf  1
6V
6<w* 5
6cfe« 14
«P/ + 1
b^cd 6
6»c« + 14
b^de +6
W/ 20
bd'e  1
c«(y10
ocfe* +6
ftc* +3
b^d' + 2
6c»«5 +1
&C(i0
cy 3
c*«»  2
(i»« 3
W(/ 26
6a^ 9
6rf» 20
c»ci(5 +9
ccTe + 26
1
c* + 12
c*rf +4
c»e +20
cd" 4
dl" 12
i 11
=k40
dbi5
=b0O
J. 15
db40
dbll
5«, y)'
J. No. 22.
o»c/« + 1
a»4/"»+ 1
a'def 2
aV/  1
aV + 1
o6c/» 2
ab*/* 1
abdef 4
aJc?/" 4
o&e' + 6
aM'/+ 8
ac"*/" + 8
tM^ a
aaPf 2
ocM^ 2
a«fe'12
w»f + 14
ad*e + 6
. acd»«  22
< ad* + 9
*»/• + 1
6»«/  2
6V + 6
6V/ + 14
Vecif\2
VdU? 16
6'c«« 16
fte»4r22
iWj +10
6«V +10
ftc*/ + 6
iedV +30
6e'<2« +30
W 16
6c«i» 20
c*/ + 9
«rt» 16
«:•«*» 20
c»d' +10
«•«<» +10
\^yY
:1:95
db95
141]
A SECOND MBMOIB UPON QUANTICS.
275
K
No. 23.
aV* ...
a\p + 1
a*e(/^ 1
aV' ••
c^cef + 1
a*e^/ 5
aV/ + 1
aMf 1
a*cy 3
aV + 4
€^P + 5
aW/+ 1
a»^ + 2
a^/« 1
cMef 8
acy«+ 3
o^V 1
ahcef + 8
a6e» + 3
(Kidef" 14
€^cdfAr 14
a6i«/+ 11
ocVH
ace* + 8
a^e»  11
aW6« 17
ocrfy+ll
ad^f + 9
oftrf"*  1
ac«(^ 11
(kcd^ + 6
ewTe*  6
ocy  9
ac»e» 16
oc^e  6
W^ " 2
. ac*flfe 4 14
V occP  6
cuid^e + 44
6»/»  4
h'def + 11
oc^*  18
ft*ce/ 4 17
6V  9
h^df  8
ft»6/  3
5«rf"/ + 16
hc'ef + 1
ft»(5» + 9
h^cdf ^ 6
h^d^  21
fccrfy  14
ft*cy + 6
6«c«> +21
6c»e(/'44
hcde" + 16
6^cei«  16
6We  6
W(5» + 5
5rf»«  3
6»rf» + 8
6cy + 6
hcd^t + 39
c*^/' + 6
6c»d + 3
h<?dt  39
5^/*  12
C»6»  8
hc'iP  2
6crf» + 22
cy + 18
c»cP« + 2
<?*(/
cV + 12
c»rftf  22
cc?*
c»rf»  8
c«(^ + 8
5^, yY
±57
±139
±129
±67
L. No. 24.
(
a^hdf ...
aV/ + 2
a'cc2e — 5
aV +3
ai^cf ^
at^de + 6
aJlx^e + 5
abed:'''!
a^d + 1
6*/ +2
6»c<!J 5
ft»(f« 2
6 Vrf + 8
6c* 3
«•/• ...
a*6^ ...
c^cdyk 7
a^ce"  10
aWtf+ 3
a6^ 7
a^«» + 10
oftcy 7
O^CG^tf— 8
rtW» + 9
oc^tf +22
ac«rf» 19
ft*c/ + 7
6»dd + 2
W«  19
bi^d +33
c» 12
aV' ...
a\?«/" 3
(^dy+12
a«<fo« 9
a6V+ 3
a6c^18
a6c6" 18
aW*e + 30
w?f  Z
00*6^ + 45
a«i» 39
hi'df • 6
6V +27
6*cy + 16
6»afo  87
6^(^ + 6
Wtf +12
6c>(i« + 57
c« 24
a*c/» 1
a^defh 7
aV  6
aiy»+ 1
abce/26
aWy+32
a6<fo« 8
ac«4/* 18
ac«e* + 6
euxPe^62
ad^ 39
l^ff +19
6«c<(/'  53
6«ce» +20
ft'ci'tf 25
h^f +39
W<fe  45
hcd^ +65
c*e
c»(P 20
a«c(r+ 1
aV/ 1
a6^ 7
abde/\'26
cM 19
oc'e/  32
acdy+18
aec;^ + 53
a^e 39
6»/« + 6
6»c«/+ 8
ft*rfy 6
6«<^20
Mdf^^h
We» +25
6crf'tf52
6rf*
c*/ +39
c»(fo 65
c>rf» +20
aV' ...
a6<(/^+ 3
oW/ 3
acy*12
ac56/'+18
occ* + 6
a^f + 3
ad'e'U
6»^«+ 9
6»^+18
6V 27
bc'ef  30
6crfy45
6ccfo*+87
c*cP«
erf*
12
+ 39
 6
57
+ 24
ay» ...
a6«/^ ...
ac4/^ 7
ace*/ h 7
a(Pe/+ 7
ode"  7
6*rf/» + 10
ft'eylO
bc'r 3
6cd;5/'+ 8
6c<5»  2
bd"/  22
6rf»e* + 19
C»6/  9
c»rf'/+19
c'de' +11
cciP« 33
<f +12
abp ...
ac«/*" ...
arfy2
ade^/ + 4
cw*  2
6'(5/* ...
6c«J/^ + 5
6cey 5
bd'ef 5
We» +5
cy> 3
C«rf6/ + 7
c«e» +2
crfy 1
cd»e» 8
ci*« +3
5«» y)'
26
93
±207
241
±241
±207
±93
±26
No. 26, Q = + 1 o»cd/' + &C.
No. 26.
35—2
276
[142
142.
NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR
ON QUANTICS.
[Now first published (1889).]
In the present paper I arrange in a more compendious form and continue to
a much greater extent the tables (first of each pair) given Nos. 35 — 39 of my
Second Memoir on Quantics, 141, pp. 260 — 264, which relate to the cubic, the quartic
and the quintic functions; and I give the like tables for the sextic, the septimic and
the octavic functions respectively. The cubic table exhibits the coefficients of the several
xz terms of the function 1t(1— ^.1— a?2r.l— a^^.l— a^z\ or, what is the same thing,
it gives the number of partitions of a given number into a given number of parts,
the parts being 0, 1, % 3, (repetitions admissible) : or again, regarding the letters
a, 6, c, d, as having the weights 0, 1, 2, 3 respectively, it shows the number of literal
terms of a given degree and given weight. And similarly for the quartic, quintic, sextic,
septimic and octavic tables respectively, the parts of course being 0, 1,... up to 4, 5,
6, 7 or 8, and the letters being a, &, ... up to e, /, g^ h or t. The extent of the
tables is as follows:
cubic table extends to deg*weight 18 — 27
quartic
quintic
sextic
septimic
octavic
it
18—36
18—45
15—45
12—42
10—40
viz. for the quintic, the sextic and the octavic functions these are the degweighta
of the highest invariants respectively. I designate the Tables as the od, a^, a/", ogi
ah' and aitables respectively.
It is to be noticed that in the several tables the lower part of each column is
for shortness omitted ; the column has to be completed by taking into it the series
142] NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICa 277
of bottom terms of each of the preceding columns: thus in the af or quintic table
the complete column for degree 3 would be
D 8
W 87
■V O
6
— 1
6
 8
5
— 8
4
— 4
3
■V n
2
— 6
1
— 7
1
where the concluding terms 2, 1, 1 are the bottom terms of the three preceding
columns respectively. And the meaning is that for degree 3, and weight 8, or 7, the
number of terms is = 6 ; for weight 7 — 1, =6, the number of terms is =» 6 ; and
similarly for weights 5, 4, 3, 2, 1, the numbers are 5, 4, 3, 2, 1, 1 ; the numbers are
those of the terms
W.
8
8
at
c?h
a'e
a*d
aU
«y
abf
acf
adf
<!&'
abe
abd
ahe
ace
ode
tuf
6'
(U?
acd
ad?
hV
be/
6'c
l?d
6V
b*ce
bde
he
bed
bcP
<?d
No.
8
The like remarks and explanations apply to the other table&
D 1 8
8
4
6
6
7
8
odTABLE.
9 10 u
18
18
14
16
16
17
18
W 818
64
6
87
9
1110
18
1418 16
1716
18
8019
81
8888
84
8686
87
1 1 2
LI.
3
3
5
4
4
6
6
5
8
8
7
7
10
10
9
8
1.
13
12
12
11
10
15 18
15 18
14 17
13 17
12 15
21
21
20
19
18
16
25
24
24
23
22
20
19
28
28
27
26
25
23
21
32
32
31
31
29
28
26
24
36
36
35
34
33
31
29
27
41
40
40
39
38
36
35
32
30
45
45
44
43
42
40
38
36
33
50
50
49
49
47
46
44
42 i
39 i
37 1
—
1
 8
1
>4
14
— 6
*
— 6
— 7
»
278 NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [142
D 12 8 4 6
6
8
(wTABLE.
10 11 la 18 14 16
16 17
18
W 8 4 6 8 10 12 14 16 18 20
24 26
84 86
113 5
8
12
18
24
33
43
55
69
86
104
126
150
177
207
241
1 1 2 4
7
11
16
23
31
41
53
67
83
102
123
147
174
204
237
~1 2 4
7
11
16
23
31
41
53
67
83
102
123
147
174
204
237
3
5
9
14
20
28
38
49
63
79
97
118
142
168
198
231
5
8
13
19
27
36
48
61
77
95
116
139
166
195
228
6
10
16
23
32
43
56
71
89
109
132
158
187
219
9
14
21
30
40
53
68
85
105
128
153
182
214
11
17
25
35
47
61
78
97
119
144
172
203
[
15
22
32
43
57
73
92
113
138
165
196
18
26
37
50
65
83
104
127
154
184
23
33
45
60
77
97
120
146
175
27
38
52
68
87
109
134
162
■
34
46
39
62
53
47
80
70
6S
101
90
82
125
113
104
153
139
129
1 54
71
64
92
83
72
116
106
93
0
1
2
8
4
6
6
7
8
9
10
U
12
U
14
16
16
17
U
a/TABLE.
D
W
1
8
8
10
11
12
13
14 16
16
17
18
82 6 87 10 1812 16 1617 20 28
26 2827 80 8882 86 8887 40 4342 46
1 1
3
6
12
20
32
1 3
6
11
19
32
2
5
11
18
30
2
4
9
16
29
3
8
14
25
6
11
23
5
9
1 7
19
16
12
10
49
48
46
43
39
35
30
26
21
17
13
73
71
70
66
63
57
52
45
40
33
28
22
18
102 141
101 141
98 137
93 134
88 127
81 121
74 111
66 103
58
50
43
35
29
23
92
83
72
63
53
45
36
30
190
188
184
178
170
161
150
139
126
114
101
89
77
66
55
46
37
252
249
247
240
233
222
212
197
184
168
154
137
123
107
94
80
68
56
47
325
322
317
309
299
286
272
256
239
220
202
183
165
146
129
112
97
82
69
57
414
414
408
402
390
379
362
346
325
306
283
262
238
217
194
174
152
134
115
99
83
70
521
518
511
501
488
472
453
433
409
385
359
333
306
280
253
228
203
180
157
137
117
100
84
649
645
641
630
619
601
583
559
536
507
480
448
419
386
356
324
295
264
237
209
185
160
139
118
101
795
791
783
770
754
734
711
684
655
623
590
554
519
482
446
409
374
339
306
273
243
214
188
162
140
119
967
966
957
948
930
912
886
860
827
795
756
719
677
637
593
553
509
469
427
389
350
315
279
248
217
190
163
141
142] NTJMBBICAL TABLES SUPPLEMENTABY TO SECOND MEMOIR ON QU ANTICS. 279
agfTABLK
D
W
18 8
8
10
11
la
18
14
16
8
6
9
la
16
18
SI
84
8T
80
88
88
89
43
46
1 1 4
8
18
32
58
94
151
227
338
480
676
920
1242
1636
1 3
8
16
32
55
94
147
227
332
480
668
920
1232
1635
1 3
7
16
30
55
90
146
221
330
471
664
907
1226
1617
3
7
14
29
51
88
139
217
319
464
648
896
1203
1601
2
5
13
25
48
81
134
205
310
446
634
870
1182
1565
4
10
23
42
76
123
196
293
431
608
847
1145
1533
3
9
19
39
68
116
182
280
408
587
813
1113
1483
6
16
32
61
103
169
258
387
553
780
1064
1435
5
12
28
52
94
152
241
359
525
737
1021
1373
10
22
46
81
139
218
335
488
699
965
1316
7
18
37
71
121
199
304
455
650
914
1244
13
31
59
107
175
278
415
607
852
1178
11
24
51
91
157
248
382
557
798
1102
19
40
78
134
222
341
512
733
1031
14
33
25
64
54
117
98
193
170
308
271
462
419
677
614
952
882
20
42
[ 34
83
67
144
124
240
206
371
331
559
499
803
734
26
56
43
103
86
180
151
289
253
449
394
661
596
(
35
69
57
129
106
216
187
349
302
529
472
44
88
70
156
132
263
223
412
362
58
108
89
192
159
303
270
71
134
109
228
195
90
161
135
110
1
3
8
4
6
6
7
8
9
10
11
18
18
14
16
16
17
18
19
80
81
84
86
87
39
80
280 NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [l42
oATABLE.
D
W
1 8
8
10
u
la
4^
7 1110
14
1817
n
86M
28
8881
86
8888
48
1 1 4
10
24
49
94
169
289
468
734
1117
1656
1 4
10
23
48
94
166
285
464
734
1109
1646
1 3
9
23
46
90
162
282
454
722
1093
1634
'
3
8
20
43
88
155
272
441
709
1069
1605
2
7
19
39
81
146
263
424
686
1038
1572
2
5
16
35
76
136
247
403
663
1000
1524
4
14
30
68
125
233
379
629
957
1475
3
11
26
61
112
214
354
598
908
1410
9
21
52
100
197
325
558
856
1346
6
17
46
87
176
297
520
799
1271
5
13
37
75
158
268
477
742
1197
10
31
63
137
239
437
682
1114
7
24
53
120
210
392
623
1036
19
42
101
184
353
563
950
14
34
86
157
311
506
871
11
26
20
70
58
134
112
274
236
449
397
788
711
15
45
36
93
75
204
171
346
300
633
564
27
61
145
256
493
21
47
37
119
98
218
182
432
372
28
78
63
152
124
320
270
48
101
229
38
80
64
189
157
«
49
127
103
81
65
11
18
18
14
16
18
17
18
19
81
84
86
88
87
89
80
NUMERICAL TABLES SUPPLBBiENTABY TO SECOND MEMOIB ON QUANTICS. 281
W
aiTABLE.
1 8
8
4
6
6
7
8
9
10
4 8
18
16
80
84
88
38
86
40
! .
1 5
1 4
13
12
33
31
73
71
151
147
289
285
526
519
910
902
1514
1502
—
— 1
1 4
12
31
70
146
282
515
894
1492
— 8
1 3
11
28
66
139
272
499
873
1460
— 8
3
10
27
63
134
263
486
851
1430
— 4
2
8
23
57
123
247
461
816
1379
— 6
2
7
21
52
116
233
440
783
1331
— 6
5
17
45
103
214
409
738
1265
— 7
4
15
40
94
197
383
696
1214
— 8
3
11
33
81
176
348
645
1127
— 9
9
28
71
158
319
597
1057
10
6
22
59
137
284
543
974
U
5
18
51
120
255
495
900
18
13
40
101
221
441
816
18
10
33
86
194
394
742
14
7
25
20
70
58
164
141
345
302
662
593
15
16
14
45
116
258
519
17
11
36
27
97
77
222
185
457
393
18
19
21
63
156
340
80
15
48
38
127
104
286
243
81
88
28
82
200
88
22
66
50
167
134
84
80
39
109
86
29
85
68
87
88
51
89
40
30
^he numbers of each table are connected in several ways with those of the
ling tables. One of these connexions, which is of some importance, is best ex
d by an example: in the a/*table, 880, the number of terms of degree 8 and
t 80 is 73 ; and we have 73 = 1 h 6 + 16 + 23 + 27. viz. (see p. 288) these
le numbers of the terms in a*, a', a", a\ a* respectively: the complementary
3, (for example) of a' are be/*, &c. terms in 6, c, d, e, f of the degree 5 and
t 80, and (replacing therein each letter by that which immediately precedes it)
are in number equal to the terms in a, 6, c, d, e of the degree o and weight
= 15 ; thus the number 6 of the terms in question is that for the deg weight
f the a6table: and so 1, 6, 16, 23, 27 are the numbers in the o^table for
egweights 4i6, 5i8, 614, 7i8 and 8i8 respectively, or (making a change rendered
lary by the abbreviated form of the tables) say for the deg weights 4o, 5io,
18 and 818.
•1
II.
36
282
[143
143.
TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUIN
TIC: FROM THE SECOND, THIRD, FIFTH, EIGHTH, NINTH
AND TENTH MEMOIRS ON QUANTICS.
[Arranged in the present form, 1889.]
The binary quintic has in all (including the quintic itself and the invariants)
23 covariants, which I have represented by the capital letters, A, B, C, . . . W (alternative
forms of two of these are denoted by Q' and S'). The covariants A, . . . L, and also
Q, Q' were given in my Second Memoir on Quantics, and except Q and Q' arc
reproduced in the present reprint thereof, 141 ; in all these I gave not only the
literal terms actually presenting themselves, but also the terms with zero coeflBcients;
in the other covariants however, or in most of them, the terms with zero coefficient*^
were omitted. It is very desirable to have in every case the complete series of literal
terms, and in the covariants as here printed they are accordingly inserted: the number
of terms is in each case known beforehand by the foregoing q/*table, 142, and any
omission is thus precluded; by means of this ci/* table we have the numbers of terms
as shown in the following list.
I have throughout (as was done in the Ninth and Tenth Memoirs) expressed the
literal terms in a slightly diflferent form from that employed in the Second Memoir:
this is done in order to show at a glance in each column the set of terms which
contain a given power of a, and in each such set the terms which contain a given
power of b.
The numerical verifications are also given not only for the entire column but for
each set of terms containing the same power of a; viz. in most cases, but not always,
the positive and negative coeflScients of a set have equal sums, which are shown by
43] TABLES OF THE COVABIANTS M TO W OF TH^: BINARY QUINTIC. 283
number with the sign ± prefixed. The verification is in some cases given in regard
I the subsets involving the same powers of a and b, here also the sums of the
)sitive and negative coefficients are not in every case equal. The cases of inequality
ill be referred to at the end of this paper.
The whole series of covariants is as follows :
Hem.
2
No. of table
13
•
A
^
(1. 1. 1, 1, 1, IJx, yy
(legweight
1 (0....5)
»»
14
B
^
(3. 3, 35ar, y)»
2 (4.6)
»
15
=
(2, 2. 3. 3. 3, 2, ijx, yf
2 (2 8)
11
16
D
ss
(6. 6, 6, 6$». yy
3 (6.. 9)
*i
17
E
=
(5. 6. 6. 6, 6, 6$«, yy
3 (5. ...10)
it
18
F
=
(3, 4. 5. 6. 6, 6, 6, 5. 4, 3$a:. y)»
3 (3 12)
1*
19
G
=
(12$a;, yy, Invt.
410
M
20
U
^
(11, 11, 12, 11, 11$^, yy
4 (8., .12)
))
21
I
ss
(9, 11. 11, 12, 11, 11, 9$a!, yy
4 (7 13)
»»
22
J
s
(20, 20$*, yy
6 (12, 13)
»»
23
E
=
(19, 20, 20, 19$«. yy
5 (11.. 14)
»»
24
L
= (16, 18, 19, 20, 20, 19, 18, 16$ir, y)'
5 (9 16)
8
83
M
=
(32, 32, 32$«, yy
6 (14 . 16)
>i
84
N
=
(30. 32. 32, 32, 30$a;, y)«
6 (13.. .17)
9
90
=
(49, 495a!, yy
7 (17. 18)
»>
91
P
=
(46, 48, 49. 49, 48, 46$*, yy
7 (15.. ..20)
2
Q 25
Q'26
Q.Q'
t
(73$«, yy, Invt.
820
9
92
R
=5
(71. 73, njx, yy
8 (19.21)
9
10
S93
S93&W
S,S'
=
(101, 102, 102, 101$ar, y)»
9 (21.. 24)
9
94
T
S
(190, 190$ar, y)'
11 (27, 28)
3
29
U
=
(252$ar, y)», Invt.
1230
9
95
V
=
(325, 326$ar, yV
13 (32, 33)
5
29a
W
=
(967$a!, y)», Invt.
1845
36—2
284
TABLES OF THE OOVARIANTS M TO W OF THE BINARY QUIMTIC. [Ud
M. No. 83.
cflfief
• ■ •
a» 6y
• • •
a* b^f*
a' 6W»
• • •
o» 6y
• • •
b'eef*
«•/
• • •
Ved^*
 1
dy  1
fiV/*
 1
c^f
+ 1
A»/ + 2
edef
+ 6
dV"
+ 1
«*  1
of
 3
d^
 1
a' ft««/»
dff
 3
a» m/*
+ 1
b'ed/* + 6
«?"«•
+ 2
«•/
 1
<!«■/ 5
a' 6»c/»
+ 2
6'c«/«
+ 1
««V  6
A/
 5
cdef
+ 6
<fc» + 6
«»
+ 3
of
 8
bvy  3
6V?/"
 5
dPf
10
c»«fc/ + 7
ccPf
+ 7
d^if
+ 11
«•«» + 2
ed<?
 1
bVef
10
aP/  1
dP«
 1
<?dff
+ 11
«?«■  8
h*(?dff
 1
<?d^
+ 18
<?« + 3
<?^
+ 6
ecPe
28
a^6«i^  3
<?dPe
 8
(P
+ 9
«!/• + 3
evf
+ 3
a<> 6»c/»
 1
6V/» + 2
a* b*/*
 1
def
 8
afo/  1
Vcrf
+ 5
«>
+ 9
«»»  3
<Pf
+ 2
l^«/
+ 11
dy + 6
d^
 3
etPf
+ 18
«?«■  4
Wdf
 8
cdt?
37
5'c««/  1
«»«■
 4
<?«
+ 8
c'd'/  8
cePs
+ 7
b'<*d/
28
cVfc* + 7
#
 1
c»«»
+ 8
wPe + 5
6"cy
+ 3
c'd'e
+ 37
<i»  3
cida
+ 6
cd^
17
b'e'd/ + 3
i?df
 4
6V/
+ 9
C«(J»  1
6V«
 3
c^
17
c»<?«  4
c^d*
 2
e'd'
+ 8
<!«# + 2
5*, y)^
:fc 7
SI
84
2
67
106
^ 2
22
28
^ 62
db 167
db 62
143]
TABLES OF THE 00VARIANT8 M TO W OF THE BINARY QUINTIC.
285
N. No. 84.
a»6«(^ 
1
a» 6»«/»
a« 6y»
a« 6y»
a« 6 V* + 1
^/ +
1
a> h^df^ ~ 4
a« 6»(5/»
6»c«/« + 4
deP  ^
1 a» h'cf^ +
3
«y + 4
6»c(^
rfy*  4
«y + 2
def +
2
6V/» + 4
c«y + 6
(i^y  4
a* 6y»  1
<5» 
5
edef  8
d««/  12
«< + 4
h'ceP  2
6V^ 
8
c«» + 4
cie» + 6
ai 6V'  4
d'P + 8
cdy +
2
d^f
a' 6"(^«  6
6*c(^« + 8
cfey  2
cd^ +
12
cP«*
ey
c«y  16
«*  6
cP« 
6
a* 6"c/» + 4
6»(y« + 12
d'ef + 48
6 V(^  2
a^fty* 
2
def + 16
cdef
dfe»  32
c»«y + 6
fe'c^/ 
2
«>  24
ce»  36
6V/«
cd^ef 20
^/ 
6
h't^ef  48
cPf + 48
c«cfe/ 40
cdfe» + 12
d^ +
13
ccP/ + 40
d«c«  12
cV + 56
rfy + 9
6ic«e(/" +
20
ccfe» + 40
6Ve/  48
cd^f + 8
rf»««  6
c«c« +
4
d»«  24
c*gP/ ...
c(?e»  40
a» 6»(5/* + 5
CflPc 
52
h^(*df  8
<?d^ + 156
rf*e + 12
6«c(^«  12
rf* +
24
c»c« + 56
ccPc  168
a» 6»(^«  4
cey  13
6V/ 
9
c>(?tf  88
rf» + 54
ey + 24
(/»«/  4
C»(fe +
20
c^ + 36
a« 6»c/»  6
6"cy»
d8» + 15
c*rf» 
10
ao 6y«  4
def + 36
cflfe/"  40
6V/« + 6
«*• 6 V +
6
h'cef + 32
o • • .
ce»  60
c«cfe/+ 52
l^cdf +
12
d«/  56
6V«/ + 12
dy  56
c«c»  10
C6« 
15
cfe> + 60
cd^f  156
d»e» + 100
cdy  20
rf*c +
10
b'i^df + 40
ccfe*
6Vc/ + 24
c(i»««  30
6V/ +
6
c>e» 100
d^e + 60
c«dy+ 88
d^e + 15
c*cfc +
30
c<?e  80
6»c»e(/" + 168
c«rfe» + 80
6Ve/  24
cd» 
20
rf* + 60
c'c'  60
ciPe  200
c»rfy+ 10
i^c^c 
15
6V/  12
c'cPe
d" + 60
c»cfe> + 20
c»rf» +
10
(*de + 200
c(^  30
6Ve(r  36
<?d^e  10
6Vrf
• • •
c«rf» 120
6V/  54
c*6»  60
ccP
6Ve  60
c*(ig + 30
c^d^e + 120
c*(? + 40
c'cP
cW  40
di
1
19
db 12
:i= 12
db 8
db 8
81
192
270
182
87
62
482
806
496
128
\^^ yY
± 168
db 686
d. 588
d= 686
db 168
286
TABLES OF THE 00 VARIANTS M TO W OF TUB BINABT QUINTIC. [143
0.
No. 90.
a» 6V/» + 1
a» b^dp  1
dtp  4
«y* + 1
«y + 3
a« 6V' + *
a« 6"/»  I
(fo/* + 3
b'ceP  3
^/ ~ 7
d'P + 16
b^t^eP  16
cfey + 4
ccr/*+ 6
«*  15
c<foy+ 30
6 V(^«  6
ce*  8
c»cy + 4
JV  18
c<i»c/ 22
d»«' + 6
cdg> + 26
a* 6»/»  3
rf*/ + 9
il'ceP  4
d»c«  12
(?/«  4
a» b^eP + 7
cfey  I
6«<^  30
tf* + 18
c^f + 1
6^(4/^+ 22
rf»c/  74
c»e»/+ 74
cfc» + 84
c<^«/160
6'cy + 18
«fo»  32
c'def + 160
rf*/ + 81
c'tf'  98
d»e» + 6
e(P/  20
fcV/*  9
cd'e'  94
(^de/^ 20
c?'^ +51
c»«» 112
6Vc/  81
c«dy 18
c»dy+ 18
c«rf»«» + 284
i^de" + 140
cd'e 216
C»d»<5  100
d» + 54
crf» + 18
a*6*c/« + 15
a» b'dP + 8
b^cdp  26
e»/  18
ce'/  84
b'c'P  6
(?»«/ + 98
crife/" + 32
d^ 45
c<5» + 45
6V/« + 12
d»/ + 112
c'de/^ 94
(i»e»  150
c««' + 150
6V«/  6
cd*/  140
c>rfy  284
ccP««  50
c»cfe> + 50
d'e + 15
ccPe + 320
b'c'ef  51
rf» 120
c>d«/ + 100
6»c*ej^ +216
c»cfe«  320
cV  15
cW« + 310
c«(i»«  310
crf»  90
e'd' + 130
6Vrf/  18
ftocy  54
c»«« + 120
c^de + 90
c*(?«  130
c*d»  40
c»rf* + 40
:&: 4
± 1
69
49
497
669
1003
954
H^. yY
1563
dbl563
8] TABLBS OF THE COVARIANTS M TO W OF THE BINABT QUINTIC.
287
P. No. 91.
• • •
a» Vf*
a»6V' 
1
a« h^df^ + 1
a» b^eP
• • •
c^ b^P
• • •
VceP  2
deP +
6
^P  1
a* bHp +
2
a^b^ep
+ 1
dy« + 6
^/ 
5
a« b'cP  6
cy« 
2
b^cdp  1
 2
d^f  1
a« 6y« +
1
d^ + n
b^^p 
5
c«y+ 1
+ 2
tf*  2
6^C<5/« 
11
ey  5
edeP^
17
(PeP+ 3
 1
a* \?eP + 2
d^P 
4
6Ve/« + 4
c^f 
7
flfey  5
 1
h^cdp  17
(^/ 
4
cd'P 2
d^P ^
4
e» + 2
+ 2
c^f + 13
c* +
17
c(iey+ 4
d^e^f
6
a} b^df^ + 2
 3
^tf  32
y^&dp^
2
cc* — 4
d^ +
5
ey«  2
 6
cfe» + 32
c»«y +
26
d'ef  10
a* 6V* +
1
h'ifp  2
+ 13
6V/« + 4
c^ef
2
rf'tf' + 8
cfo/« 
13
cdtp^ 6
 8
c»cfe/+ 36
cd^ 
40
a' 6y» + 5
^/ +
12
cey  2
+ 2
c»«»  24
rfy 
9
b^cep + 4
6'cV^ +
32
d^P  16
+ 16
cd^f  10
rPe' +
24
(^/«  26
cd^P^
36
d«cy + 24
 2
oi»«»  16
a} h'ep +
5
de"/  35
cd^f
42
rfe*  10
38
d^e + 12
6'crf/« 
4
«< + 42
ce* +
24
6V<5/« + 8
+ 34
a> hHp + 7
ct^f +
35
b'c^dp + 2
cP^ +
56
(?d^P^ 2
 9
ey  12
rf*?/" 
26
cV/+ 26
cP«» 
34
c'cfey 52 !
+ 5
6V/« + 6
efo« 
22
cgPc/" + 72
6V^« +
10
cV + 28
+ 2
cdef + 42
6'cy +
10
cde"  124
c»cy
54
cd?ef ^ 52 j
 12
C6^ • • •
c'def
72
rfy + 13
c^d^ef+
64
ccPe^  32
24
e^f + 54
cV 
106
d"^ + 26
c^d^ +
46
d^f  18
+ 52
(f«e«  91
ccP/ +
76
6^cy* + 9
cd'f 
37
rfV + 12 1
6Vc/  68
ccPc* +
210
cW«/ 76
cd^e^ 
50
a« 6V' + 1 '
22
c»rfy_ 64
d^'e 
99
cV  56
cPe +
21
ei^/«  13 i
52
c«cfe« + 14
¥c*ef 
13
c«c^/+ 10
a« 6y» +
2
ey + 12
+ 34
ccPc + 204
i?d^f
10
c^cTe^ + 296
6»c^/» 
32
b^c^eP  2
+ 8
^  93
i^de" +
128
cd'e 260
c^y^ +
24
ccP/ + 38
 1
V'i^df + 37
c'c^e 
184
c^» + 72
d^f
• • ■
crf«y 7 1
+ 18
cV + 86
cc? +
72
a' b'ep  17
e*
• • •
ce*  30
25
c*(^c  208
a« 6*c(/^ +
4
6»c4/'« + 40
6V(y« +
16
d^ef  34
+ 10
c«rf^ + 86
^/ 
42
cey + 22
c^^y +
91
d^^ +35
 2
a« 6^c/«  5
6V/« 
8
d"^ + 106
cd^ef
14
6Ve(/^  34
+ 10
cfe/  12
cd«/* +
124
(ie^ 105
cd^ 
105
c»ey + 22
28
c • • •
cc* +
105
6V/«  24
d^f 
86
c«rf«c/ 8
+ 30
6Ve/ + 34
rfy +
56
c«(fe/210
cPc« +
110
i?df? + 50
+ 32
crfy  46
C^6» 
130
c»c» + 130
6V/« 
12
ccfy + 25
35
cd^ + 105
6V«/ 
26
cd"/  128
i^d^f
204
cdJ'^  70
50
c^e  20
<?d^f
296
cd"^ + 170
i?^ +
20
rf*c + 15 1
+ 30
l^i?df + 50
i?d^ 
170
rf^e  25
(^d^f +
208
6V/« + 9
 12
c»c« 110
ccPe +
340
b'c*e/ + 99
c«c^e« +
170
c*<fe/+ 1
+ 70
c*ci»e  170
c/» 
60
c»rfy + 184
cc?*c 
250
cV 30
40
cd?' + 115
h'i^df +
260
c»c£e«  340
^ +
60
c'ciy  10
15
6V/  21
cV +
25
c'd/'e + 150
6Ve/ +
93
c'(^c« + 40
+ 10
d'dA + 250
c'cPe 
150
ccf  40
c'd^f
86
<?d^e  10
> • a
c«rf»  150
c«c^
« • ■
b'c'df  72
^d^ 
115
C€^
6V«  60
6«cy 
72
c»c« + 60
c^cPc +
150
&d^ + 40
c"cfe +
40
c*d^e
cW 
40
•
c^d^
« • •
c'd*
± 3
=k= 6
Jt^
6
^ 1
67
99
70
27
±
24
± 6
136
536
536
577
266
134
182
594
954
961
«
944
248
5«, uf
^ 388
dbl234
:tl566
rtl566
±1234
db38a
288
TABLES OF THE CO VARIANTS M TO W OF THE BIKABT QUINTIC. [l43
Q. No. 25. Q'. No. 26.
Q. No. 25. Q'. No. 26.
a« bof*
• • •
+ 1
a» 6V
+ 27
 3375
a* b'ef*
• • •
20
h^t^dp
 48
1 5760
b'cd/*
+ 1
 120
<?i?f
+ 3
 600
c<?P
 1
+ 160
cd^ef
+ 106
 16000
(Pep
 3
+ 360
ed^
 81
+ 9000
d^f
+ 5
 640
d^f
 38
+ 6400
«•
 2
+ 256
rf»c«
+ 38
 4000
a* b'df*
 1
+ 160
6V/«
+ 18
 2160
«•/*
+ 1
10
i^def
 30
+ 7200
W/»
 3
+ 360
&^
+ 38
 4000
cdeP
+ 11
 1640
t^d^f
+ 8
 3200
<^f
 5
+ 320
<?d^^
+ 25
+ 2000
d»/«
+ 12
 1440
cd^e
 57
• • •
c?^f
 30
+ 4080
rf»
+ 18
<fe«
+ 15
 1920
b'i^ef
 9
6V«/»
+ 12
 1440
c*d*/
+ 6
<?cPp
 21
+ 2640
C*(fo«
 57
<?d<?f
 34
+ 4480
c'd'e
+ 74
<?^
+ 22
 2560
c'd*
 24
c^rf
+ 78
 10080
y'cHf
• • •
edP^
 48
+ 5760
c«c«
+ 18
^f
 27
+ 3456
c^cf'c
 24
<?«»
+ 18
 2160
c*rf*
+ 8
a' VcP
+ 5
 640
d<P
 5
+ 320
«y
. • •
 180
l^<?ep
 30
+ 4080
eePP
 34
+ 4480
edff
+ 133
 14920
ct^
 54
+ 7200
«P«/
 18
+ 960
dP^
+ 3
~ 600
b^i*dp
+ 78
 10080
e^f
 18
+ 960
<?d^ef
 220
+ 28480
<»d<?
+ 106
 16000
cd>f
+ 93
 11520
c<i*«s*
 30
+ 7200
Thesm
018 for Q' are
d*e
 9
• * .
1
= 1
b'>(*P
c*def
 27
+ 93
+ 3456
 11520
776
21266
68656
 780= 4
 21250= +6
 68660= 4
«!*«•
 38
+ 6400
87816
 87815= +1
<?d^f
c'rf's*
— 4.9
+ 5120
 3200
+ 8
128505
128505=
c'd'e
+ 6
• • •
cd"
. • •
• • •
a* 6»/»
 2
+ 256
6V^
+ 15
+ 1920
«P/»
+ 22
 2560
d^f
 54
+ 7200
± 6
til 128505
169
525
424
^1124
3] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC.
289
R.
No. 92.
• • •
a» Vc^f  15
a* 6y*
• • •
a^ 6V
• • •
o» jy
• » •
a« 6»d»e/ + 2
• a •
rf«e/ 38
a' 6 V*
• • •
6Vc^ +
18
Veep
• • .
dJ'd' + 15
• • •
cfe* + 46
h'cdp
• • •
i^^f 
66
dy« +
1
6Vrf/« 32
 1
6»cy« + 3
c^P
• • •
cd^tf ■\
20
d4?P
2
c»e»/  39
+ 6
c«(fe/+ 102
(^e/« +
2
cd^
• • •
«y +
1
cWe/ 24
 4
c»«»  16
cfey 
4
d*f +
58
a* 6V'
• • •
c«dfe» + 175
 3
cdy+ 76
6» +
2
d'e' 
50
l^cdp 
6
c^/ + 25
+ 1
crfV  175
a» }^dp
• • •
6V/' 
6
c^P +
6
cd»c«  120
+ 1
d^e + 35
^P
• • •
c^def h
72
cPeP +
3
c^e + 15
+ 2
6*cV  42
h^i?P 
2
cV +
50
d4?f
• • •
6V/« + 9
 6
c»(^/~ 182
cdeP
• • •
c^d^f
156
«• 
3
d'def + 106
+ 4
i?d^ + 120
ct^f +
4
c«rf*e»
• • •
6V/' +
3
c*e»  35
 3
c>d»« + 150
d»/» 
14
cd^e +
90
<?deP
3
c»dy 60
+ 3
c(i»  70
d^^f +
30
d» 
30
<?«*/
6
c»d«e»150
 18
h^&df + 126
d^ 
18
6V^/ 
24
cePp
• • •
c»rf*c + 176
+ 17
&^  15
6Vc/« +
14
cW +
94
cd^^fh
3
crf«  45
+ 22
c*d»«  175
cW/«
• • •
i^d^ 
90
c<fo* +
6
6Vc/  36
 21
(?d* 4 75
(^d^f
66
c»^e
• • •
#«/
• • •
d'd^f^ 21
• • •
6V/  27
(?^ +
26
c«c?» +
10
<?«• 
3
c»rf6> + 70
+ 13
c«(fe + 45
cc^ef +
56
h^d'df 
18
a' 6'd/» +
4
c*d»«  75
 12
c»(^  20
cd^^ 
18
C«6» +
30
«•/' 
4
c>d" + 20
 21
^f 
18
<^d^e ~
10
6V/» 
1
 3
d^^ +
6
c^c^^
• • •
c<fo/^ +
18
+ 32
a» 6»c/« +
4
w*/ 
16
 9
dep 
4
ePP 
13
 1
^/
• • •
cP^/
3
• • •
6 V«/« 
30
d^ +
15
+ 6
cd^P^
66
h^<?fp 
22
+ 16
cde^f
• • •
c'cPP*
12
 18
c^ 
18
(?d4?f+
18
 3
d^ef 
84
c*e* +
38
+ 3
d^iS" +
66
C€Pef +
32
 18
h'i^dP
56
cdP^ 
102
+ U
c^^f ^
84
<Pf 
18
 41
^d^tf
% » *
d**" +
42
+ 39
c^d^ 
20
Ve^dp +
3
• • •
cd^f +
40
cV/ +
41
 32
cd^^ 
72
<?cPef
84
 2
(Pe +
24
<?d^ 
76
+ 84
6V/« +
18
c'd'/ +
33
+ 24
c*de/
40
«•<?«• +
182
106
c^e» 
58
cd»« 
126
+ 36
c'dy
• • •
rf' +
27
+ 18
c'cPe' +
156
a'iV* 
1
~ 33
± 8
c«c^e 
94
± 4
deP 
17
:&: 2
 25
98
cc?* +
18
136
«y +
18
21
+ 60
300
a' b'P 
2
476
6Ve/» +
21
465
 21
780
6*c«/^ +
18
478
cd>/« +
21
693
+ 3
±1181
d'P 
26
:tl094
cd^f
14
:lrll81
 6
d^f +
18
ce* —
45
\«^ y)'
C. II.
37
290
TABLES OF THE C0VABIANT8 M TO W OF THE BINABY QUINTIC. [l43
S. No. 93 6m ; S'. No. 93. (•5«. y)».
CJoof. *•
S
•
3'
Coef.x»
S
8'
Coef. j:*//
s
8'
Coef. £*y
8
V
a* by*
• • •
a} l^d'ef
 66
+ 528
a* 6V*
+
9
a> b't?^
+ 6&
Vcef*
+
9
d^e"
+ 72
45
deP
—
45
C(Pef
+ 7B\
cpr
+
21
h^(?dp
 21
 2592
^P
+
36
c^^
 1&&
d^r
—
78
C»6>/
 96
 9747
a» 6y*
—
9
d^f
c*/
+
48
ed^ef
+ 36
 8496
6W*
—
18
d*««
V
cfVeP
—
9
^d^
+ 213
+ 26610
d^P
+
243
b'd'dp
+ L ^*
h^cdp
—
162
cd^f
+ 120
+ 8544
d^P
+
9
cV/
__ ^^
c<?P
+
99
C(P^
 303
 16650
eV
—
216
<?d^tf
—
dPtP
+
309
d^e
+ 51
+ 720
V'&dp
 3
—
351
C»d8»
—
A*/
+
12
h'^P
+ 9
+ 972
i^^P
+ 3
+
144
c*d^f
—
«•
—
240
^def
+ 174
+ 24624
cdPeP
+ 24
+
1836
i^d^i?
+ 1 
6V/»
—
2
—
81
cV
 36
 5040
cd^f
 42
—
2592
cd^e
+
<?der
+
15
+
1026
i^d^f
 204
 15984
C6»
+ 18
+
1152
d'
—
<!»«•/
—
9
—
768
c»e?«»
 174
 29340
d^P
 18
—
1458
W/«
—
eeP/*
—
9
—
738
(^d^e
+ 330
+ 34320
(Pt?f
+ 33
+
2268
<^def
+
«?«»/
—
6
—
664
cd^
 99
 8640
(i»«*
 15
—
1008
c««»
+
ede*
+
9
+
1056
6V(5/
 63
 7776
a^h*ip
• • •
+
63
<M^f
+
d*ef
+
9
+
756
^d^f
+ 66
+ 6184
b'cdp
+ 6
—
234
cV?««
•
<?«»
—
7
—
696
<^d4^
+ 99
+ 12960
cep
 6
—
18
<?d^
+
o?Vdp
• • •
+
120
d^e
 147
 14400
d*ep
 24
—
3231
c'd'
+
«•/•
• • •
—
21
c>d»
+ 45
+ 3840
^f
+ 42
+
4293
a* ft»c/»
+ _
v<»p
+
6
+
486
a* 6y •
+ 2
+ 192
^
 18
—
972
deP
— ^^
edef*
—
30
—
2160
h^ceP
 15
 1440
h^i?P
+ 3
+
810
^f
+ tfS
c^f
+
18
+
1023
d^r
 6
^ 192
<^dep
 78
—
3826
h^eP
.^ ^
d}/*
+
9
+
120
*y
 18
 1080
<?i?f
+ 69
+
4032
cdy*
+ Sf
d^f
+
6
—
1053
<5*
+ 27
+ 2025
cd^P
+ 93
+
7938
cd^f
 9^
(fe«
—
9
+
1314
b'c'tif*
+ 24
+ 1728
cd^^f
 61
— .
9360
c«*
 zr'
h^eP
—
15
—
1863
i?^f
+ 61
+ 4410
ede'
 33
—
864
d^ef
^r 60*
<?€Pr
+
21
+
2538
cd^ef
+ 102
+ 5280
d>ef
 57
—
1296
(?«»
 45
<?def
—
6
+
2340
cd^
 171
 13500
(Pe"
+ 54
+
2700
h^<^dp
 39V
<^e*
+
18
+
672
^f
+ 6
 4800
h\^eP
+ 24
—
324
t^^f
+ 45
'^
ed^ef
+
30
+
2820
d^i?
+ 18
+ 7800
i?d^P
 36
—
2484
i^d^tf
 108
«?«•
—
61
—
7812
l^c'P
 9
 648
i^d^f
 9
+
6624
c>cfe»
+ 96
r
dy
—
36
—
3024
&def
 210
 14040
c»«*
 64
—
6912
cdyf
 Ill
^
d***
+
39
+
4672
<*^
+ 43
+ 3076
i^d^tf
+ 24
—
4428
edP^
+ 147
—
JVt^'
—
3
—
324
<?d^f
 120
+ 9120
c>d»«>
+ 129
+
12672
d^t
 30
+
c'e'/
+
46
+
3888
C>(?«'
+ 345
+ 16350
cd^f
+ 9
+
1944
6V/«
+ 9
+
i^d'ef
—
84
—
8748
cd^t
 87
 19200
cc?*e'
 114
—
9072
cW
+ 6
—
c'de*
—
63
—
4800
d"
 2
+ 4800
d^e
+ 27
+
1944
<?*«>
 48
—
^d^f
+
45
+
4248
IMef
+ 72
+ 4860
a^h^dp
 3
+
144
i^d^f
+ 234

<?dPt?
+
150
+
14620
d'd^f
+ 240
 3240
^P
+ 3
—
243
i?d^^
 150
»
(xPe
—
117
—
11448
(H^
 192
 8100
b'i^P
 6
—
900
C\^J
 108

cP
+
27
+
2692
(*d^e
 186
+ 9000
edeP
+ 108
+
10620
erf*
+ 57
a> b*e/*
—
6
—
576
c»d»
+ 96
 2400
c/f
 96
—
8586
h^^tf
+ 9
def*
+
15
+
672
h^d'df
 144
d^P
~ 21
—
864
i?d^f
 141
«•/
—
9
—
459
C«6>
+ 18
d^^f
 48
—
1215
&d^
+ 87
V<?eP
+
30
+
3466
i^(Pe
+ 201
df^
+ 63
+
1215
e^e
+ 96
cd^r
—
15
—
864
c'd^
 87
6V^
 24
—
1836
i^d"
 51
ed^f
+
24
+
2094
h'<?f
+ 27
(?d^P
 123
—
16812
b^e'd/
+ 27
C9*
45
3915
i?dt
i^d^
 46
+ 20
cW/
+ 147
+
6651
 18
 21
+ 12
For the Namerical Verifications for S see
further pp. 304, 806.
78
db as 3258
414 41253
1284 124524
1292 68640
db8028 :k 287758
± 78
480
927
96 6
db245i
143] TABLES OF THE OOVARIANTS M TO W OF THE BINARY QITINTia
291
S. No. 93 Us; S'. No. 93.
f«
s
8'
Coef. xy'
S
8'
Ck>ef. y*
S
S'
Coef. y»
S
8'
• • •
_
9
a^b'i?^
_^
60
+
4320
a* 6 V*
• • a
a^ b^d'd
 72
 4860
1
• • •
+
9
i^ef
+
36
+
14544
a» b'd/'
—
9
b^'deP
+ 36
+ 3024
• •
+
45
i^dPi^
+
108
—
3060
^P
+
9
dd^P
~ 45
 4248
t
• • •
+
18
c^f
—
24
—
5184
¥df*
—
21
dddf
 120
 8544
1
• • •
—
63
cd"^
—
6
+
1620
cdep
+
162
dd
 6
+ 4800
•B
• • •
—
243
d'e
—
9
• • •
cdp
—
120
dd^ef
+ 204
+ 15984
r»
+ 3
+
351
h^<*dp
—
9
—
1944
rfy»
+ 2
+
81
d(Pd
+ 120
 9120
/•
 6
+
234
i^i^f
—
51
+
3888
d}dp
 6
—
486
dd^f
 66
 5184
■
+ 3
—
144
d'd^ef
+
96
—
1296
de*f
+ 6
+
576
dd'd
 240
+ 3240*
1
 3
—
810
C*d8»
+
111
—
1440
d
 2
—
192
cd^e
+ 144
• • •
f
+ 6
+
900
f?df
—
27
+
576
a« h'cP
• • •
+
78
<f»
 27
• • •
 3
■—
288
c»d»««
—
234
+
360
dtp
• • •
—
99
d' b^ep
• • •
+ 240
• • •
—
36
c«d»«
+
141
• • •
^P
• • •
+
21
b^cdP
 9
 1056
1
• • •
—
9
erf'
—
27
• • •
h^deP
• • •
—
309
cdp
+ 9
 1314
1
 3
—
144
a^l^dp
—
18
—
1152
cd^P
 15
—
1026
d^eP
 18
 672
ri
+ 6
+
18
^P
+
18
+
972
cddp
+ 30
+
2160
ddf
+ 45
+ 3915
 3
+
243
6V/«
+
15
+
1008
cdf
 15
—
672
d
 27
 2025
1
 24
1836
cdep
+
33
+
864
d'ep
+ 15
+
1863
l^dp
+ 7
+ 696
n
+ 24
+
3231
c^f
—
63
—
1215
^df
 30
—
3456
ddeP
+ 51
+ 7812
r
+ 78
+
3825
(PP
+
54
+
6912
dd
+ 15
+
1440
ddf
 72
+ 45
f
 108
—
10620
d^^f
—
66
—
12960
V'ddp
+ 9
+
738
cd^P
+ 63
+ 4800
+ 30
+
3888
de'
+
27
+
6075
ddp
 9
—
120
cd^df
 213
 26610
1
 24
+
324
6V«/»
—
54
—
2700
dd'eP
 21
—
2538
cdd
+ 171
+ 13500
f
+ 24
+
1836
i»d^P
—
129
—
12672
dddf
+ 15
+
864
d^ef
+ 36
+ 5040
• • •
—
756
C'd4?f
+
186
+
18900
dd
+ 6
+
192
(Pd
~ 43
 3075
1
+ 18
+
1458
c>«*
+
45
—
6075
cd^p
+ 3
4
324
l^deP
 39
 4572
/'
 93
—
7938
cd^ef
+
54
+
12960
cd'df
+ 21
+
2592
dd'p
 150
 14520
f
+ 21
+
864
cd^(^
—
96
—
10125
cd^d
 24
—
1728
dddf
+ 303
+ 16650
r
+ 36
+
2484
d^f
~
54
—
5760
d^ef
 9
^
972
dd
 18
 7800
«y
+ 123
+
16812
rf*e»
+
48
+
4500
d^d
+ 9
+
648
dd^ef
+ 174
+ 29340
4
 51
—
7488
l^^P
+
114
+
9072
a' b*P
• • •
—
48
dd*d
 345
 16350
f
 Ill
—
15228
c*«y
+
9
—
2970
b^ceP
• • •
—
12
c^f
 99
 12960
t
+ 39
+
7128
f?d^ef
—
150
22500
d'P
+ 9
+
768
cd'd
+ 192
+ 8100
+ 27
+
.3888
dd^
—
147
+
13950
ddP
 18
—
1023
d^e
 18
• • •
 9
—
1944
dd^f
+
93
+
9360
ey
+ 9
+
459
b'ddp
+ 117
+ 11448
• • •
+
216
d^d
+
150
—
6300
l^ddp
+ 6
+
564
ddf
 51
 720
1
+ 42
+
2592
ed^e
—
87
• • •
ddp
 6
+
1053
dd'ef
 330
 34320
•a
 42
—
4293
d?
+
18
• • •
cd^eP
+ 6
—
2340
ddd
+ 87
+ 19200
1
 69
—
4032
b'dp
—
27
—
1944
cddf
 24
—
2094
dd^f
+ 147
+ 14400
•
+ 96
+
8586
ddef
—
30
+
6480
cd
+ 18
+
1080
dd/'d
+ 186
 9000
 27
—
3645
dd
+
30
—
3600
d>p
 45
—
3888
dd^e
 201
• • •
 33
—
2268
dd^f
—
6
—
2880
d^df
+ 96
+
9747
ccT
+ 45
• • •
/^
+ 51
+
9360
d^d
+
108
+
1800
d'd
 51
—
4410
b'dp
 27
 2592
r
+ 48
+
1215
dd^e
—
96
• • •
b'dp
 9
—
756
ddef
+ 99
+ 8640
rt
+ 9
—
6624
dc?
+
21
ddep
 30
—
2820
dd
+ 2
 4800
y
 147
—
6651
b^def
+
27
• ddf
+ 66
—
528
dd^f
 45
 3840
+ 39
+
4050
dcPf
—
9
ddJ'P
+ 84
+
8748
dd^d
 96
+ 2400
•
+ 78
+
4968
ddd
—
57
dd^df
 36
+
8496
dd^e
+ 87
• • •
 45
—
2970
dd^e
+
51
ddd
 102
—
5280
dd^
 20
• • •
•1
+ 57
+
1296
d(P
—
12
cd^ef
 174
—
24624
/•
 24
+
4428
• cd^d
+ 210
+
14040
y
 78
—
18612
d^f + 63
+
7776
He 9
=k 12
1548
426
45999
912
62019
1101
92853
=b 8
db 828
123
10920
1071
79779
1821
146226
:fe2451 ±202428
±8023 ±237753
37—2
292
TABLES OF THB OOVABIAMTS M TO W OF THE BINAirr QUINTIC. [14
(
T. No. 94.
X coe£Scient
a coefficient
a* VcP
a'lt'^P
_
20
a> ft«c»d'«/»
+
153
a« 6V/« 
6
def*
d'^P
+
33
<?d^f
—
390
c»cfe/« +
240
«•/'
<foy
—
48
cV
—
234
«•</■ +
179
a* bV
«•
+
27
ed*P
—
114
c»d»/« 
144
b'cef*
V&fp
+
39
cd*<?f
—
308
c'cPe'/ +
306
<P/*
<?d}p
—
105
c«P«*
+
735
c'd^ 
765
de>P
<?d^P
+
18
d^ef
+
208
cct^f +
28
e*/*
<?t^f
—
6
d*e»
—
283
cd"«» +
280
VtMf*
 "i
ed?eP
+
114
wp
+
27
<p/ 
88
<?^P
+ 1
edPtff
~
67
<*deP
—
396
(PeF +
40
ccPef*
+ 7
edtf
+
12
c'e'/
—
337
6"c»«/« 
63
cd4?p
 12
dPp
—
6
<?d^P
+
222
c\iPP +
42
c^f
+ 5
d'e'/
+
3
<^<P^/
+
783
<?*cfe>/ 
798
#/>
 6
d*»^
—
12
c«<fo«
+
880
cV +
176
cP^P
+ 12
l^d'dP
+
90
c'd^ef
+
93
d^iPef 
224
dfi^f
 7
<!««•/»
—
198
e'd'e*
—
1986
c»rf»«» +
1366
d^
+ 1
(?d}eP
—
9
cdff
—
240
cW/ +
368
c? i'e/"
• • •
<?d^f
+
238
cd>^
+
1098
c»d*«« 
1026
Vcdp
+ 2
<!»«•
+
116
d'e
—
144
cd^e +
60
c^P
 2
^dfp
—
6
b'ifcP
+
81
d» +
30
6?tP
 7
i?d*^f
+
108
<fd'P
—
54
l^^dp
• « •
d<?P
+ 12
c**?**
—
613
(fde'f
+
570
if^f +
252
«•/
 6
cdPef
—
294
c»««
—
148
c»c?«/ +
798
6'cy«
+ 3
edV
+
513
e*(Pe/
—
1116
C»d8» 
700
<?deP
 30
d'/
+
108
c*d'e*
—
627
cV/ 
578
i?^P
+ 21
(?«•
—
153
c«d»/
+
474
iN^i? 
370
ecPP
+ 44
bfd'p
—
27
<?d>^
+
1662
c'cPtf +
880
edP«?P
 69
<?deP
+
108
<?d^e
—
1185
<NP 
240
«fey
+ 62
c»«'/
+
194
eeP
+
243
6V/«
...
c««
 28
«*</•/*
—
42
iVdp
• • •
<?def 
486
d^eP
 6
c**?**/
—
663
c'ey
—
216
<?^ +
60
d»e'/
 8
<*dif^
—
274
(fd'f/
+
369
«•(/»/ +
312
<?«•
+ 11
ed^tf
+
570
d'd^
+
340
c«dV +
645
6Ve/»
 6
€»<?«•
+
914
(I'd*/
—
149
(*d^e 
735
ifd^P
 11
<?dff
—
163
(!»(?«"
—
730
e^ +
190
<?d^P
+ 96
cW«»
—
1032
C*(P«
+
488
6V«/ +
81
<?eV
 64
ccPe
+
486
e'd'
—
102
C»(i»/ 
54
<?df«P
 66
d*
—
81
a' VP
—
2
c»cfe» 
136
<?(P<?f
 29
a» VeP
+
7
Veep
+
20
(?d^e +
150
<?dtf
+ 68
deP
—
16
d^P
—
24
c»d» 
40
ecPp
+ 18
^P
+
9
d^P
+
72
ed^f
+ 76
b\?eP
—
53
•
«y
—
64
cd^(^
 78
cd?P
+
104
v<?dp
+
16
^ 26
d^ef
 27
cd<?P
—
150
c"«y
—
129
486
a* b*dp
+ 24
 1
d}iP
+
117
48
cd'ef*
ed^f
+
108
72
8788
9116
6880
v<*p
+ 1
d^ff
df
+
138
€«■
+
135
1 ^
 8
1
108
d*P
+
84
:!: 90196
edeP
+ 46
v&dp
—
82
d'e'/
—
112
and see farther
p. 806.
c^P
 30
c»«y
+
316
d»««
• • •
143] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC.
293
T. No. 94.
y coefficient.
y coefficient.
t^b'df'
a« 6"crf»ey«  18
a' b^d'dp 
75
a<>6^jd»ey 
880
</•*
cd^f + 150
<^^P 
3
ccPtf* +
765
a* hhp
c««  72
&d^eP 
108
d^ef +
148
def^
dl'ep + 198
<?df?f +
308
^•«' 
175
ey
d»ey  315
c»«» +
112
6V/» 
24
h^(?ef^
rf««» + 129
<?d^P +
663
d'dep 
513
cd^f
+
1
h'd'eP + 6
c«rf»ey 
783
cV/ +
283
cd^P
—
2
c8rf»/» + 66
• c»rf«e* 
306
i^d^P ~
914
c^P
+
1
<*d^p  114
ccJ'e/' 
570
c»rf»ey +
1986
^fp
—
3
c>«y + 48
cd^^ +
798
<*d^ 
280
d^^P
+
8
c^d^ep + 9
^/ +
216
t^d'ef +
527
d^f
—
7
c«rf»ey  153
(i»«»
252
c«rf»e» 
1365
^
+
2
(?d^ + 108
6^cy» +
27
cd^f 
340
(^¥p
• • •
cd^P  108
C'flfe/^ +
294
cd^^ +
700
b'cep
• • •
crf»cy + 396
c»ey 
208
cTe 
60
d^P
—
1
cd^^ ~ 240
c'd^^f 
93
6»c*e/« +
153
d^P
+
2
d:'ef  81
cV/> 
570
^d^P +
1032
^r
—
1
d"«» + 63
c^cfo* 
28
(?d^f 
1098
h^i?dp
—
7
6V(^»  18
c*c?'e/' +
1116
c»«* 
40
(?^p
+
7
c»cy> + 6
c'c^c* +
224
i^d^ef 
1662
cd'ep
+
30
c^d^ep + 6
c»dy 
369
c^d^^ +
1025
cd^P
—
46
d'd^f + 114
c«d»c« 
798
c'rfy +
730
c^f
+
16
c*«»  84
cd?e +
486
c»(^e» +
370
d^P
+
6
<?d^p + 42
d»
81
c^cJ'e —
645
(Pe'P
—
39
<?d^(?f  222
6Vfl/« 
108
c^ +
135
iP^f
+
53
c>rf»«* + 144
i^d^P +
153
6^c'(y» 
486
A»
—
20
(?d^ef + 54
c«rfey +
240
cV/ +
144
h^p
+
6
c"rf»«»  42
c««* +
88
d^d^ef +
1185
edrf*
—
44
ccPf
<^d?ef
474
c«(ie» 
60
e^p
+
20
cd^^
i^d^^ 
368
dd^f 
488
<?d^p
+
11
d^e
d'cPf +
149
dd^d 
880
c'd'^P +
105
a* mp  5
c^d^^ +
578
dd^e +
735
c'd^^/
—
104
ey* + 5
c'cJ'e —
312
dd^ 
150
^^
+
24
b'i^P + 7
0*6? +
54
6V/« +
81
I cd^eP
—
90
edep  62
a« 6 V*
1
ddef 
243
cd^^f
+
82
c(^P + 48
cfe/» +
28
dd
30
C(?<J»
—
16
(PP + 64
^P 
27
dd^f +
102
d^p
+
27
d'e'P + 6
fc»C«(5/» 
11
dd^d +
240
d^^f
—
27
efey  117
cd'P 
68
c'c?*e —
190
d^^
+
6
«• +54
c(iey« 
12
c'c?' +
40
a* 6V*
• • •
• 6Ve/» + 8
C€*/ +
108
l^cdp
+
12
c'd'P + 29
d^eP 
116
e^P
—
12
i^d^P + 57
d'e^f +
234
i 12
d}tp
—
21
c>«y  138
cfo»
135
895
d^P
+
30
cfPeP  238
6Vci/^ +
78
1650
//
—
9
ceP«»/ + 390
c»6y +
12
6511
4 *« £^£^£\
^<?P
+
12
69
cd^  72
rfy«  194
i^d^eP +
513
735
11628
i?dep
=1=20196
f?^P
—
33
c^«y + 337
c»«»
...
and Bee farther
p. 806.
\ ed^p
—
96
d»^  179
cc^/« +
274
5*. y)'
294
TAPLW OF THB C0VABIANT8 M TO W OF THK BINABT QUINTIC. [14
U. No. 29.
a* b"/*
. a* J'ePa* 
22
a« 6V«i»/»  108
a'Wi^P 
90
o» VdPP 
a* 6 V»
6V/« 
4
cW/ 42
AP^ 
42
dV/ 
6»«i^'
<!*</«/» +
36
<!»«?«« + 298
<W/ +
674
d»«* +
«!«•/*
^^P 
16
odV" + 242
<!♦«•
4
b*<*dp +
tPef*
(?tPP 
22
«iV  294
c»d'/» +
394
<ft?P +
d^f*
ifd^P
50
d»/  72
c»dV 
662
<?d^P +
*P
ifd^f +
16
«?«• + 78
<?<P^/
714
(?dif/ 
t^Vf*
<!»«• +
16
V'^dp  6
<?cPnf 
498
<!»«• +
Vdf*
c»«?e/» +
54
*•«•/» + 62
c'dV +
1246
<*d^P +
^f*
e^iff +
46
c'dV  108
ed'/ +
224
<?^^f 
jvy*
<!•«?«• 
60
fl»«fc»/  164
od»«« 
516
<?^P^ +
cdef*
t^P 
6
c»«»  24
d"* +
48
afef +
tPP
ccfeV 
70
cy/* + 63
6V/» +
18
cdV 
<?«•/»
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56
cV««/ + 394
i'ds/' +
242
dPf +
d^P
rf'e/ +
18
cy»«« + 194
cV/ 
128
<P(j» +
«•/
«<•«• 
14
edftf  324
o'dy 
324
6»c»/» 
6V«/«
.. a» ft*/*
« • a
c»dV  440
c'dV/ 
498
(*d^ ~
c«<?/« 
1 6««/«
• • •
c'd'/ + 78
c»d(J* +
136
&^f +
c'de»/» +
2 «?/« 
1
«»<?«• + 428
««V" +
1078
e<d»/« 
c»«y» 
1 Ay* +
2
«*•«  180
(A?** +
206
cW/ +
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6 «y» 
1
6P» + 27
c»dy 
342
c«d«» 
cJP^P  1
6 W4^« 
16
a> 6V*
c'd»(j» 
804
c'dV +
«W/ + 1
4 «V/» +
16
ft»«(^« + 14
(Me +
506
c'd»e« 
ce'
4 tdPeP +
82
e^P  14
cd»
90
cV/ 
<?/• 
4 cefe»/« 
132
<P^  32
6V«/» 
72
c^V +
(^e*/* + 1
1 c^f +
50
d«»/» + 50
e'tPP +
78
eJPe
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««•/• 
16
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224
d»
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3 df^P 
14
bVp  10
<?<^ +
16
6«c'5r +
o» 6»^»
dV/ +
60
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(?tP^ 
342
c»d'/» +
de/*
d* 
30
c»«»/» + 60
(fdf* 
220
C«d9»/ 
«•/•
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11
edVp 48
<fd>f +
106
c»«« +
6V«/«
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30
c«iy + 16
<fd>^ +
392
eepif 
cd»/' +
2 c»«y« 
14
ed^/ + 88
cV« 
222
c»dV +
«W/» 
4 cW/' 
60
<!<•  86
<?6P +
40
cW/ +
c«y» +
2 <!»<?«•/• +
168
d*ep + 112
a« Vdp 
4
c«V +
d'«/» 
6 «!**•/ 
48
d**"/  204
t?P +
4
c»d«« 
^^p + 1
6 c»<C 
4
(Pif + 102
6V/« +
3
cV +
rf**/  1
4 cdV* 
48
6»c«</^ + 50
edrf* +
24
^ +
4 «?«•/ 
2
c»d»/» + 46
<?d?p  2
e»««/  204
©«•/» 
SO
6'c"<(/*« +
6 «P«» +
6
d»/» +
16
ed}«f +
«!•«•/» 
6 #/» +
62
d»e»/' 
4
c'de» 
«!»</•«/»  5
rfe*/ 
90
<?(P^  170
d»«/ 
36
efd*/ 
c'd8'/»+ 8
2 d»«* +
39
«!**• + 308
if +
27
c«d»«» 
oV/  3
2 iVe/* 
28
cV**/ + 42
i'cdV* 
104
c*d»« +
cd>P + 3
6 c<d»/» +
64
«?/>  164
c»^ 
22
cW 
«?«•/•  3
<*d^p 
48
cdV/ + 674
c»d»/» 
60
6V«/» +
crf»««/  3
cV/ +
112
etP^  590
<fdt?p +
6
«•*/ 
e(i«* +2
4 c»d'^ +
82
dV"  128
cV/ +
102
c»e»
dftP  2
.8 c»d»«'/ 
170
(i>«« + 138
cdW/ +
308
«^d»/ +
«W/ + 5
<*d<f 
104
W4f*  70
cde* 
234
c'dV +
(Cd* +
rkSe, ^464, db2608, :k7378, d:6878, together :&: 17264: and see farther p. 807.
143] TABLBS OF THE COVABIANTS M TO W OF THfi BINAftY OUtNtlC.
iu
V. No. 95. (•$«, y)».
» ooeffioient.
0^ 6»c/*
<f b*t^'
^
2
a» 6*orf»«'
+
876
a« 6Vrf"<^
+
2800
d^
^f*
+
2
d^f
+
162
i^d^rf
+
6624
«•/*
V(?P
—
16
rf»fl»
—
162
i^d^^
+
2052
«• 6y«
cdeP
+
32
a« }^rf^
+
14
c^f
—
918
6'c«/»
c^P
• • •
def^
—
6
ccTe*
—
2304
(Pf
tPp
—
8
«•/•
—
8
d»«
+
486
«fay*
«?«•/»
+
80
b'c'ef'
—
50
c'dp
• • •
«•/*
d^P
—
160
ed"/'
+
90
h'(?i?P
+
504
V<?df*
—
2
</•
+
72
cd^P
—
120
i^d^p
—
576
<(•«»/<
+
2
ve«p
+
84
c^r
+
60
i^d^f
—
2288
«?«/*
+
10
^dfp
—
104
^rf^
—
280
i^4^
+
1172
ed^P
—
16
ifd^P
—
160
d^^P
+
300
<*d*p
—
124
ofP
+
6
«;•««/•
+
60
d^f
+
216
c»rf»«y
+
4336
itf*
—
6
cefep
+
320
^
—
216
dd^f^
—
2540
^^f*
+
12
et^P
+
80
h'i^df^
—
160
d'd^ef
—
1912
dV/»
—
10
odff
—
496
&^P
—
80
c*d*«»
+
2100
d^f
+
6
eg
+
252
^d^P
+
1280
^d?f
+
240
«•
—
2
dPp
—
72
<?d^P
• a •
&d^^
—
1560
a* 4«^
• • •
dVp
—
420
c»</"
—
312
&d^€
+
810
Vedf*
+
4
ePe*/
+
860
cd^P
—
440
ecP^
—
162
c^f*
—
4
dV
—
404
cd^e^P
— •
2160
a» 6y»
—
4
d^ef*
—
10
b'^dp
+
96
edS^f
+
1740
h^CBP
1
22
«&•/»
+
16
cV/'
—
120
cd^
—
216
d'P
~
26
«•/•
—
6
i?d?^P
—
560
d^tp
+
2344
dd'P
+
76
WP
+
6
«?dgp
+
160
d^ef
3240
^P
• • •
^def*
—
26
«»«•/
+
304
rf»«»
+
1244
}^i?dp
+
124
«•«•/•
+
8
<?d^P
+
280
WP
+
72
f^^P
+
368
ed^f*
+
32
^dfgp
+
1440
(MeP
—
240
cdFep
—
688
«?«•/»
—
116
<?<Pe*f
—
900
&i?P
+
940
cd^P
—
192
tdi^P
+
180
e'dtf

376
f?d^P
• • «
^f
• • •
«*•/
—
78
ed^eP
—
1296
&d^^P
—
1320
d^P
+
400
<ft./»
+
24
ed'g/
+
80
&df^f
—
2640
d^^P
+
984
<?«!•/»
—
20
cd^g
+
832
fl»^
+
908
d^^f
—
2160
««•«•/
—
44
«r/»
+
432
i^dSp
+
600
d^
+
1080
dg
+
34
dfgf
—
72
&d^^f
+
3360
h^d'p
—
60
6»«V*
—
30
dP^
—
240
(?d^f^
—
168
^dep
—
480
^d^f*
+
4
Wp
—
36
ed^P
—
1656
&^p
—
1580
ifd^fp
+
240
ifdtp
+
288
fid^^f
+
3408
i?d?P
+
40
«•«*/»
—
130
(fgp
—
56
ed"^
—
3480
f^d^^P
+
2040
^^»P
—
160
<^d}p
—
140
d?ef
—
1008
i^dt^f
+
2910
(?(Pt?P
—
280
(fdP^P
—
480
d^i^
+
1224
c»^
—
810
^d^f
+
332
iNitff
+
420
h'ifep
—
144
6d^€f^
—
3420
«»«'
—
54
«?•<*•
—
276
&d^P
+
108
cdFi^f
+
4800
efiP
+
24
ed^«P
+
420
^d^P
—
768
ed"^
—
3510
td^P
+
360
edfgf
—
1120
&^f
—
700
d^p
—
1516
«?«♦/
—
320
&d'f
+
1112
d^eP
+
900
(P^f
+
2156
edV
+
38
cW/»
—
144
f^d^f
+
8160
d^
—
430
dfiP
—
108
i?*^f
+
1620
C*(fo»
—
2148
W^
+
336
#«•/
+
96
(MV
—
1620
&6^P
+
912
d'd^P
—
40
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—
12
ed'ef
—
864
^d^^f
—
15060
f^d^p
+
2640
For the Nomerioftl Yerifioations see p. 806.
296 TABLES OP THE COVAMANTS M TO W OF THE BINABT QUINTIC. [143
V. No. 95 (continued).
X coefficient.
a' i'c'e*/
+
1840
a" Vd^p
+
•
184
a» 6*c»«/«
 594
«•<?«/»
—
1280
d^P
■
108
ed^p
 10296
<?^i?f
—
13360
«v
• • •
(?d^f
+ 10080
c»<W
+
3200
v<?p
+
18
(?^
+ 900
<?dfP
+
7312
i?deP
+
264
&d?^
+ 19440
^d^ff
—
2360
<?^P
+
756
C«(?«»
 8800
<?dP^
+
3840
c^P
—
368
f^iPf
 9160
cd^^
—
5344
cd^^P
—
732
i^d^
11900
ed*(?
+
2800
od^f
+
540
c*d»«
+ 13900
rfy
+
1956
ce«
• • •
c»rf»
 3150
iPt?
—
1680
d^p
—
1172
h^&dtp
+ 3564
6Vd/'
—
36
df^f
+
2520
c»«y
 1350
«•«•/»
—
1296
rf***
—
1350
c»dV
 9540
<?dPeP
+
1668
6'c*e/»
—
144
i^d^
 750
<*dtf/
—
1312
^d?p
+
376
ed^f
+ 4260
«•«•
—
2060
<?d4?p
—
1440
dd^i?
+ 10800
cV/*
—
8020
(f^/
—
1530
c«d»c
 9100
cy«»/
+
15220
cW«/«
+
6360
<*d?
+ 2000
c«dV
+
1180
<»d^<?/
—
6000
b^c'P
 486
<»eP^
+
3712
i?d^
+
1350
^'def
+ 1620
ifd'tf

8540
cd?P
+
2344
c^V
+ 450
^d^f
—
2952
ed*^/
—
9260
&d^f
 720
<!•<?«»
• • •
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+
7200
&d^
 2250
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+
3330
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+
1720
d'd'e
+ 1800
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—
810
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—
1900
dd"
 400
wp
• • •
h\*d{p
—
168
e'd'p
—
576
«•«•/•
+
648
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+
1824
t^tNp
~
6420
i?d?p
+
3792
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+
9360
c»d'«y
—
5808
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+
450
tfd^
+
3240
«•#/»
—
10100
<fd^/
—
4768
&dfff
+
19920
<*d?^
—
6240
<?dW
—
10300
(^dPf
+
2608
cWe/
+
4920
c*^<?
+
12440
cW
—
10100
<?(Pe
—
8160
cdff
—
3440
<^cP
+
1620
cdV
+
7100
b'<»eP
+
162
d»«
—
750
(i'd'P
~
702
wp
+
36
c»rf«»/
—
90
ifdip
+
2988
c«««
—
1290
&i?f
—
2880
c'dV
+
1920
<fdfp
+
14688
c'dV
+
3640
<fd^i?/
—
22740
<fd'/
—
796
<*d^
+
600
&d^
—
5340
<*d^ef
—
16520
fl»d»«
+
3100
<!«(?«•
+
23300
<j^d»
—
600
edff
+
8760
a* Vip
+
18
«•<?«'
—
5200
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—
36
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—
5400
m^P
—
180
<5rf»
+
1500
143j TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC.
297
V. No. 95 (continued).
y coefficient.
«• fdf*
a» V'<?deP
„«•
116
a« 6 V/
• • •
a^ 6Vrf*ey
+
7312
_. **/'
e'^P
+
80
6Vc/*
—
20
i^d^
+
2344
«• *'«/•
cd'P
+
240
<^<Pf'
—
280
i^d^P
—
124
dtf*
eePe'P
—
160
<?d^p
+
80
c*^ey
—
8020
^r
ed^P
—
120
i?e^p
+
300
c*d^
—
10100
*Ve/»
«•/
+
76
cd?ep
+
160
C»(f»«/
+
3792
«P/»
—
2
tfep
—
120
edJ'^P
• • •
c>rf»«»
+
14648
«fcy*
+
4
d^e^P
—
80
cd^f
—
192
<?d^f
—
702
a^P
—
2
(P^f
+
368
c^
—
108
&d?^
—
10296
dfef*
+
6
dg
—
180
<PP
—
56
cd^e
+
3564
df<?P
—
16
V<*ep
+
24
d}^P
+
940
J"
—
486
d^P
+
14
<^cPp
—
160
d^e'f
—
1580
a^ 6V'
+
6
* ^f
—
4
<*d^P
+
320
d"^
+
756
deP
—
78
<»* iy«
• • •
<?eP
—
280
6Vrf/*
+
360
^P
+
72
*•«/»
• • •
^d^eP
—
560
cV/»
—
420
h^(?eP
—
44
d»/'
+
2
ff^^P
+
1280
&d^eP
+
1440
cdPP
+
332
d^P
—
4
i?d<ff
—
688
(?d^P
—
2160
cd^P
—
496
«y
+
2
i?g
+
184
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+
984
c^P
+
216
b'c'dp
+
10
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+
288
c^d^p
—
480
d^eP
+
304
e^P
—
10
cd*^P
—
240
&d^eP
—
1320
d'i^P
—
312
e<PeP
—
26
ccPey
—
480
(?d^e^f
+
2040
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• • •
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+
32
«?«•
+
264
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—
732
^
• • •
c^P
—
6
dPtP
—
144
cd^ep
—
768
b*(^dp
—
320
d'P
—
30
d^<?f
+
336
cd*^f
+
2640
<?^P
+
860
d*^P
+
84
d*^
—
144
cd^^
—
1440
i^d^eP
—
960
df(^P
—
50
b'i^dp
+
24
d^P
+
504
&d^P
+
1740
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—
22
<f^P
—
72
d^^f
—
1296
<?^f
—
2160
«»
+
18
d'cPeP
+
280
d^e''
+
648
cd'P
+
420
wp
—
6
(fd^P
—
440
b'(^P
—
108
cd^i^P
—
2640
&d»P
+
32
C*«»/
+
400
d'dep
—
1296
cd^^f
+
2910
e^P
—
8
&d*P
—
140
i?^P
+
2344
cd(^
+
540
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+
4
^df^P
• • •
d'd^P
+
420
d^eP
—
700
<M}i?P
—
104
c»d«ey
+
40
i^d'^P
+
600
d^f
+
1840
<?dt^P
+
90
c»<fe«
—
368
<^d/f
—
3420
d^
—
1530
c"**/
—
26
c"d»«/»
+
108
d"^
—
1172
h^^P
+
96
cd*eP
+
96
<?d'^f
—
40
&d^ep
+
900
d'dep
+
80
edf^P
—
160
c»d»«»
+
376
i^d^f
—
1280
f^^P
—
3240
«?«•/
+
124
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• • •
^d!"^
+
6360
&dp
—
1120
C(*^
—
36
c^^f
—
36
i?d^P
—
576
&d^P
+
3360
dFp
—
36
cd?<^
—
168
i^de^f
+
1668
ifde^f
+
4800
df^P
+
72
dPef
• • •
c«rfV
—
6420
c>««
+
2520
d>t^f
—
60
d?t?
+
36
cdef
.—
576
d'dep
+
8160
__ <?«•
+
18
o« 6»<^»
+
6
cd^e^
+
2988
^d^f
—
13360
**• h'ep
• • •
«y'
—
6
d^f
+
162
i?d^
—
6000
Vcdp
—
16
6V/»
—
10
d»(5"
—
594
cdp
—
2288
c^P
+
16
cdrf*
+
180
b^<FeP
+
432
cdt^f
—
1312
d?ep
+
8
c^P
—
160
d'd^P
—
144
cd*^
+
9360
d^P
• • •
tpp
—
130
(^d^P
—
1656
def
+
1824
«•/'
—
8
d»ey>
+
60
cV/
—
1516
d^
—
2880
6V/»
+
12
dt*P
+
60
&d^eP
+
912
h^d'eP
—
72
For the Nmnerical Yerifioatioxis see p. 808.
C. II.
38
298
TABLES OF THE OOVARIANTS M TO W OF THE BINABT QUINTIC. [143
V. No. 95 (concluded).
y coefficient.
a^ b^d'cPf*
+
1620
a« h^P
^^
276
a« 6*c«<fo*
+ 7100
&der
+
3408
d^t^P
+
908
i*d^ef
+ 12440
&^f
+
2156
d}^f
—
810
cd***
 5200
c'cPep
—
15060
d^
...
<N^f
 5340
c^d^^f
—
2360
6»cy*
—
12
c'tPe'
 11900
c*cfe»
—
9260
<^d^'
+
832
&d?t
+ 10800
i?d'P
+
4336
&i?P
■♦•
1244
c«d»
 2250
i?d^^f
+
15220
i^d^P
+
1112
h'd'ep
+ 486
(*<Pt^
+
19920
c'd't^P
•
168
i?d}p
+ 810
d'd^ef
—
5808
<?de'f
—
3510
i^d^f
+ 3330
<?iJ^f?
—
22740
c«e«
—
1350
c»«*
 750
cd^f
—
90
cd^fp
—
2148
ed^tf
 8160
Cd^i?
+
10080
cd^^f
+
3200
ed^i?
 5400
dl'e
—
1350
cd'tfi
+
1350
d^d^f
+ 3100
h^edp
—
864
d^P
+
1172
i^d^t?
+ 13900
e^P
—
1008
d»«»/
—
2060
<*d^e
 9100
<^d?ep
+
6624
c^«^
+
450
<jW
+ 1800
i^d^f
—
5344
h'&eP
—
240
h^i^^dp
 162
i^i?
+
1720
<^d}p
—
1620
e^^i^f
 810
&d>P
—
1912
c'd^P
—
3480
(^d}ef
+ 1620
&^i?f
+
3712
<?*</•
—
430
&df?
+ 1500
c»cr»«^
+
4920
<*d^ep
+
2800
i^d^f
 600
i^d^tf
—
4768
i?d^i^f
+
3840
d^d^t?
 3150
d'd^
—
16520
&d^
+
7200
dd^e
+ 2000
&d^f
+
1920
<?d^P
—
2540
i^d?
 400
C«(?«»
+
19440
i^d^i^f
+
1180
c»d"«
—
9540
c»cP«*
—
10300
cdy"
+
1620
Cd^if
+
3240
6V/»
+
162
cd»«»
+
600
^dep
—
918
d^f
—
1290
i»^f
+
1956
d?i?
+
900
ed^p
+
240
V't^dp
+
876
(?d^^f
—
2952
ceV
+
1224
edi^
—
3440
<^d^^P
+
2052
d^d^ef
+
2608
&di?f
+
2800
i*d^^
+
8760
&^
—
1900
&d^f
—
796
^d^P
+
2100
&d^i?
—
9160
d^d^i^f
—
8540
c^d'e
+
4260
c*cf«*
—
10100
c»rf»
—
720
(?d^ef
—
6240
a' bp
—
2
i^d'i?
+
23300
l/cef*
+
34
i?d!f
+
3640
d"/*
—
54
e^i?
—
8800
d^P
+
252
c<Pe
—
750
eP
—
216
d}'
+
450
h^c'dp
+
38
WP
—
162
i?^P
—
404
t?deP
—
2304
cd'eP
—
376
t?^f
—
1680
cd^p
—
216
i^d^P
—
1560
c^f
+
1080
&d^^f
« • •
143] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC.
299
W, 29 a.
a»6«/»
• « •
o^ h^c^P
 1
5 a» 6Vdey* 
90
a» 6V(£«fiy
+ 1320
a»6'e/«
• • •
d^^P
+ 1<
cV/» +
30
<?d^
 260
b'cdp
« « •
(t^P
 3.
5 cd'ep 
210
f?d?p
+ 60
c^r
• • •
iP^P
+ 4(
ctPe'P +
120
d'd^^P
 500
cPeP
• • •
d^^P
 1<
cd'^p +
360
d'd'ep
+ 2235
<&»/•
• • •
dA?f
 li
cd^P 
420
(^d*d'f
 1995
«•/•
• • •
^
+
5 cc»/ +
130
i?d^^
+ 370
a' If dp
• • •
h^i^P
—
1 d^P 
5
cd^ep
+ 360
^r
• • •
c^deP
+ 1.
5 rfV/* +
195
cd?^P
 1320
1fi»p
• • «
c'i^P
 K
a d'eY* 
315
Cf^^f
+ 1110
edeP
• • •
&d?P
• •
d^d'P +
40
cd^^
 210
e^P
• • •
c'cPe'P
 9<
rf»ey +
165
d^P
 81
tPp
• • •
(^dk^P
+ 12<
d^^ 
75
d^^P
+ 270
«?«•/•
• • •
i?^P
 4<
6V€/« 
10
d^^f
 225
dt/'P
• • •
i?d'tP
+ 61
c*rf»/« 
60
cfe«
+ 45
«•/'
• • «
f^d^t^p
+ 31
D i^d^p +
210
a* U'eP
veep
• ■ •
f?d^^P
 18
c*«y* 
110
h^cdp
tftpp
• • •
(?dep
+ 12
i?d^ep
« • •
ceP
<?di?P
• • ■
<?i?f
 2
f^d'^p 4
60
d^eP
<*e*P
• ■ •
cd^P
 1
5 i?d&p 
360
d^P
ed^eP
• • •
c^^p'
 11
<?^p +
240
^r
cd'^P
• • •
cd^eP
+ 26
5 c«rfy» +
30
h'i^p
 10
cd^p
• • •
cd'^P
 20
i?d^^p 
210
i?deP
+ 90
cey
• • •
cd^/f
+ 6
5 i^^ep 
180
d'^P
 60
d^P +
1
cde'^
 1
&d}^p +
1140
cd^P
 120
d^^P 
5
d?eP
+ 4
5 i^d^f 
870
cd^i?P
+ 90
dPe*p +
10
d^e'P
 10
i?^^ +
130
cdt^P
• • «
ePe*/' 
10
d^i^P
+ 8
1 cd^ep +
310
c^P
• • •
<fc»/» +
5
d^e'f
 3
cd^d'P 
240
d'ep
+ 110
«"/ 
1
rf»«»
+
5 cd^i^p 
390
d^^P
 50
cfVcP
• • •
a» h^p
ccPey +
280
d^^P
 240
dtp
• • «
h'ceP
ccPe* +
30
d^P
+ 280
ey
• ■ •
d?p
d^P 
180
^f
 90
V<?eP
• • •
d^P
(fey» +
300
h^i^eP
+ 35
cd'P
• • •
eP
dJ'^p 
120
i?d^P
 30
ed^P
• • •
h^i^dp
d^^f +
30
(^de'P
 120
ofP
• • •
i?^P
rf*e»
30
<?tP
+ 50
<?«/•
« • «
cd'eP
\P(^dp +
15
<?d^eP
 60
dPt»p
• • •
cd^P
&^P +
5
i?d^i?P
« • •
d^P
• • •
cd'P
&d^eP 
30
i^d^P
+ 360
^r
• • •
dp
+ i
&d^p 
270
i^dp
 210
\f<*dp
• • •
d^^P
 4
&^p +
196
cd^P
+ 270
c'e'/'
■ • •
d^^P
+ 6
c^d^p
« • •
cd^^P
+ 575
<»d*eP
• • •
d^P
 4
c^d^^p +
225
ccPtp
 1700
<?d^P
• • •
^P
+ 1
(^d^tp +
615
cd^^P
+ 480
i?^P
• • •
h'd'P
+
5 i^d^P 
660
cd^f
+ 670
cd*P 
15
c'deP
 6
c*ey +
45
ce"
 315
cd?<?P +
60
c't^P
+ 4
i^d'ep 
120
d^eP
 685
ccPey* 
90
i?^P
+ 9
&d^^p 
220
d^^P
+ 540
««*?•/» +
60
c'd'e'p
•
i?d?^P 
980
d^i^P
+ 1515
For the Numerioal Verifications see p. 809.
38—2
300
TABLES OF THE COVARIANTS M TO W OF THE BIXART QUINTIC.
[143
W, 29 A (contmued).
a* 6»<i'«y
2080
a* 6^C(^^
^^^
615
a»6»rf»/»
__
196
a^6*(?e«
 3880
9f
+
705
d}^ep
—
945
dV/«
—
660
h^dep
 300
b^(*df*
+
110
d^f
+
900
dP^r
+
1840
i^d^p
+ 500
<^^r
—
195
d"«»
+
45
<?«•/»
—
1040
d'd^P
+ 3810
d'iPef^
+
210
h^d^ep
+
180
d^f
—
180
i?^p
 3710
i^d^P
—
575
edp
—
60
«»»»
+
216
ed^tp
 14040
i^i^P
+
660
ed^p
—
1420
h*<^df*
—
265
&d^^P
+ 16120
c'd'f^
—
225
cV/»
+
25
<?♦«•/•
+
315
&d^P
 540
i^J^^P
+
1350
i^d^ep
+
780
<*(Pef*
+
180
c»ey
+ 600
^d^^P
• • •
i^d^^p
+
5760
<»d^P
+
1700
d'd^P
+ 7020
i^d^P
—
1440
i*d^P
—
2945
(!»«•/•
—
1840
i^d't^P
• • •
c»ey
—
75
(^ey
+
1390
<?d>P
—
615
d'd^e^P
 1950
(?d^ep
—
1965
&d^P
...
t?<Pi?/^
—
1350
i^d^^f
 17670
<?d^^P
+
6000
&d^^P
—
7020
^d?/f*
• • •
C*(fo»
+ 4170
<?d^^P
^
7050
&d^tP
—
180
c'tfc'/*
+
1560
&d^tp
+ 480
i?d^^f
+
3000
&d}iff
—
1275
<?*/
+
135
&^^P
 31040
i?d^
+
265
&d^
—
1110
cd^ef*
+
2210
ed^^f
+ 45180
cd^P
+
1420
c^d^eP
+
3120
cd}<^P
—
4100
<*d^d
 3160
cd'^P
— .
3810
d'd^d'P
+
3900
ccPfp
+
6000
f?d^P
 140
cd^i^P
+
2310
(^cy
+
1240
ccP^f
—
4880
i^d'^P
+ 18000
cd^^f
+
1795
cV«'
+
3155
oM
+
990
i^d^^f
 12180
cd^^
_^
1800
i?d^P
—
515
<r/«
—
25
<?d^i^
 13430
^eP
+
240
i*d^P
—
2920
d^/*
+
3710
cd^eP
 7200
d^d'P
+
30
&d^e^f
—
940
d««y*
—
10755
cd^^f
 120
d^^f
__
870
c«d»e«
—
4300
d^f
+
9875
cd?^
+ 9960
d^^
+
615
^d^ep
+
675
<?«•
—
2845
d'P
+ 1890
h^eP
_
45
ed^^f
+
510
6V/«
+
100
rf^v/
 540
i^dtp
_
310
i?d'^
+
2940
&def*
+
240
d»e*
 1710
i?^P
+
685
cd>P
...
<!•«•/*
—
540
b'ddp
 360
&d^P
+
120
cd'^'e'f
—
135
<*d}f*
+
220
de'P
 240
i^d'e^P
+
1965
cd^e'
—
990
<^d>^f*
—
6000
dd^P
+ 5840
&deP
—
2210
rfiy
• • •
e*d6*P
+
4100
dd^P
 6560
&^P
—
960
c^V
+
135
c***/*
+
1340
ddp
+ 8460
e'd'ep
• • «
c^h'dp
• • •
ifd^P
4
11700
dd^P
 3120
c'd^d^P
—
11700
^P
• • •
t^dP^P
• • •
d^dp
 480
i^d^^P
+
15435
h^i?P
+
10
<*d^^P
—
15240
dd^dP
 25880
i^d^f
—
2760
CileP
—
60
edgf
+
6960
dddf
 1820
d"^
+
555
c^P
+
40
&i?
—
1620
dd
 3620
<?d^P
—
780
d^P
+
40
<»d*/*
—
5760
dd^eP
+ 49680
&d^^P
+
14040
d^^P
—
30
c»d'«y»
—
16120
dd'dp
« « •
i?d^tp
—
10625
de'p
• • •
c'd'e'p
+
26700
dd^df
+ 17520
i^dl'^f
—
3220
^P
• • «
<?d'<?/
—
5240
dd'd
+ 13500
&d}^
^
570
h^&eP
—
40
c'rf"*'
—
1640
dd^P
 120
(^deP
—
5840
i^d'P
+
180
ed?ef*
+
6560
dd^dp
 32280
i?d^^P
—
540
f^d^P
—
360
od*^r
+
7240
dd^df
 46880
&d^^f
+
5550
c»«y*
+
240
cd^ff
—
24240
dd^d
 30040
c*rfV
+
1285
cd^eP
+
360
ed^d
+
11420
dd^eP
+ 12860
cd^P
+
990
cd^^P
—
360
d^f*
—
980
dd^df
+ 32000
cd^^p
+
3150
cd^P
• • •
(P^P
—
3420
dd'd
+ 46160
cdH^f
—
3600
c^P
• • •
d?^f
+
8100
dd'^P
 2700
143] TABLES OP THE COVARIANTS M TO W OF THE BINARY QUINTIC.
301
W, 29 A (continued).
«» b^<NPi?f
_
8820
a» l^cd^^p
+
1440
a» h^cd^d
^^
18750
a« h^dd?d
+
243000
C>(i»«*
—
34620
cd^d'P
—
1560
d^ep
+
14115
dd'^ef
+
2340
cd^tf
+
1080
ed^P
• • •
d^df
—
23790
dd^d
—
89550
cd^t?
+
12060
c^f
• • •
d?d
+
8175
cd^f
+
270
d^f
• • •
d^tp
+
960
Wdp
+
1320
cd>^d
+
15120
d^e"
—
1620
d^^P
—
1340
ddp
—
30
d^e
—
810
b^c'T
+
81
d^f^P
—
2440
dd^ep
+
540
b'd'ep
+
945
edep
—
990
d^df
+
4320
dddp
—
7240
dd^p
—
675
f?^P
+
980
dt^
—
1620
ddp
—
20390
dddp
+
7200
^d^P
+
515
\^&P
—
81
dd^P
—
3900
ddp
—
14115
f?d?^P
+
UO
d'dep
+
390
dd^dp
+
31040
dd^eP
—
12860
^d^p
—
195
d'dp
—
1515
dd^dp
+
32370
dd^dp
+
8220
c»ey
—
5575
dd^P
+
980
dddf
+
38820
dddf
+
150
^d^tP
+
120
dd^e^P
+
7050
dd
+
9310
dd
+
6155
ed^^p
—
800
ddtP
—
6000
dd^ep
—
49680
dd^P
—
480
dd^i^f
+
22600
d^P
+
2440
dd*dp
• • •
dd'dp
+
26700
ddd
+
7240
d^ep
—
15435
dd^df
—
91260
dd^df
+
63960
f^d^p
• • ■
ddPdP
+
15240
dd^d
—
50550
dd^d
—
6660
f^d^^p
—
1260
dd^dp
• • •
dd^P
+
800
dd^ep
• • •
d^d^/f
—
42330
dddf
—
6480
dd^dp
+
81840
dd^df
—
180600
c«^e«
—
34340
dd
+
1215
dd^df
+
360
dd^d
—
71610
f?d'tp
+
480
ed^P
+
2945
dd*d
+
101450
dd^P
—
4755
ffd^^f
+
48360
cd^dp
+
540
dd^ep
—
8220
dd'df
+
141240
C»(i»C»
+
73828
ed^dp
—
795
dd^df
—
58080
dd^d
+
219730
c*(iy»
+
105
cd^df
—
4180
dd^d
—
34300
dd^ef
—
45130
c^d^f
—
30265
cd}d
+
4185
cd}p
—
7590
dd^d
—
240975
C*(f<?*
—
92290
d^ep
—
8460
cd^df
+
41640
dd^f
+
5580
ed>^ef
+
9540
d^dp
+
20390
cd^d
—
4650
dd'^d
+
128490
c»(i»e»
+
69220
dPdf
—
16194
d}^ef
—
5580
dd^e
—
34155
c»(P/
—
1215
d^d
+
3765
d}^d
+
1980
cd"*
+
3645
c»rf"««
—
30510
h^dep
+
120
b'dp
—
270
h^d^dp
• • •
crf»«
+
7290
dd^P
—
2235
ddep
—
3150
d'dp
—
1890
—
729
dddp
—
2310
ddp
+
3420
d'd*ep
+
2700
^^\?cP
—
5
ddp
+
10755
dd^P
+
2920
d^ddp
+
7590
deP
+
15
dd'eP
+
10625
dd^dp
—
18000
d^df
+
8256
/^'
—
10
dd^dp
—
26700
dddp
+
43800
dd'P
—
105
IPc^P
+
10
dddp
+
795
ddf
+
5030
dd'dp
—
14360
ed'P
—
120
ddf
—
10070
dd^eP
+
32280
dd'df
—
43605
cd^P
+
420
dd^P
+
180
dd^dp
—
81840
ddd
—
12310
c^p
—
280
dd^dp
+
1950
dd^df
—
85800
dd^ep
+
4755
d^ep
—
240
ddPdp
• • •
ddd
—
28710
dd'df
+
77790
d^^p
+
210
dd^df
+
36510
dd^P
+
1260
dd^d
+
59835
df?P
• • •
ddd
• • •
dd^df^
« • •
diC^f^
• > •
Jf"
• • •
dd^ep
+
25880
dd'df
+
181980
dd^df
—
57060
\^edp
+
200
dd^dp
—
32370
dd^d
+
153480
dd^d
—
114960
&^p
—
40
dd^df
—
12180
dd^ep
—
26700
dd^ef
+
19020
^d^ep
—
1140
ddFd
—
9850
dd^df
—
41360
dd^d
+
109660
^deP
—
480
cd^P
+
195
dd^d
—
306900
dd}^f
—
2481
e^p
+
1040
ed'dp
—
43800
dd^P
+
14360
dd^d
—
56110
cd^P
+
660
cd^df
+
72755
dd^df
—
16170
dd}^t
+
14895
302
TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. [143
W, 29 A (continued).
a» 6 VeP
_^_
1620
a> b^d^P
+
5575
a^ h'cd^^
+
41250
a> b^<^P
+
5580 I
a' b'^f
+
1
d'e'P
—
5030
d^f
+
5445
c'^d^P
—
9540 \
b^ce/^
+
10
d^t'f
—
4255
d}^^ .
—
6525
c'^d'ey*
2340 1
d^P
+
20
d^^
+
2175
}^<?ep
—
900
i^'^d^f
+
20610 1
d^P
—
130
b^i*dp
—
1110
<?d}p
—
510
c^V
—
4350 1
^p
+
90
cV/*
+
870
(»dep
+
120
<?d^P
+
45130 1
b^i^ap
—
65
&d^ep
—
5550
^t'P
+
23790
d'd^d'f
—
92200
c»ey»
—
165
<*df?P
+
24240
ed^tp
—
32000
&d^d^
—
25050
cd^eP
+
870
&^P
+
16194
f?d^^p
+
58080
e^P
—
19020
cd^P
—
670
d'd'P
t
1240
^d^f
—
15440
i^d^^f
+
46050
Ci^P
+
180
i^d^^P
—
45180
cV
—
12500
cWj*
+
138750
d^P
—
45
d'd't^P
+
12180
<^d^P
—
48360
dd!ef
• • •
d^e^P
+
75
i^d^f
—
66650
i^d'i^P
+
41360
dd^i?
—
178200
d^e^P
—
135
cV
—
8550
d'd^^f
—
181600
^d?f
—
1650
d^P
• • •
&d^eP
—
17520
c»^^
—
18400
c»cf^«»
+
103950
ey
• • •
<?d'^p
+
91260
&d^eP
+
180600
&d}^t
—
30250
b'c'P
+
30
c'd^e^f
• • •
&d^i?f
• « •
c*cP
+
3600
<?deP
—
280
c'd'e'
+
62100
c»c^e»
+
289800
b^i^eP
• • •
<?i?P
+
2080
i^d?p
—
22600
c'd^P
—
77790
d^d'P
+
1215
&d?P
—
1320
i^d^^P
+
85800
i^i?f
—
87000
d^d^P
—
270
i?d^^P
—
3000
i^d^e'f
—
148890
c*rf««*
—
318500
d^f
—
5445
i^de^P
+
4880
c'd*^
+
1850
<?d^ef
+
92200
c^'cPeP
—
5580
i^i^P
—
4320
cd^ep
—
150
c'd'^
+
179500
c"cW/
+
17520
cd^eP
+
2760
cd^d'f
+
15440
c'd''/
—
17520
c"cfo»
+
8700
cd^^P
—
6960
cd^^
+
10350
c'd'^^
—
69000
d'd^p
+
2481
cd^^P
+
6480
^ioy2
—
8256
cd^e
+
15300
d^d^i?f
—
10595
cd^f
• • «
<^^/
+
12210
d"*
—
1350
<^d^^
—
31150
c^
• • «
c/»c^
—
7050
b^c'^dp
+
135
dd^tf
+
1650
d^P
—
1390
b'^P
+
225
c^^^P
+
540
i^d^d
+
37950
d^^P
—
600
c'deP
+
3600
&d^ep
+
8820
dd^f
• • •
d*^P
+
10070
c'^P
—
8100
(?d^P
—
41640
ddPt?
—
22275
d^i^f
—
12600
(^d^P
+
940
i^f^f
—
12210
ddf'e
+
6600
c^e»
+
4050
i^d^^P
+
12180
i»d*P
+
30265
dd^
—
800
b^^ep
—
30
d'ilep
—
72755
&d?^P
+
16170
a« &"«/•
—
5
i^d^P
+
1995
c«.y
+
4255
i^d^e'f
+
62025
b'^edp
+
10
d'd^P
—
1795
(*d*ep
+
. 46880
d'dt^
+
44225
edp
+
75
c^eV
—
9875
<^(P^P
—
360
dd^ep
—
141240
d^fP
—
130
i c»c/»c/*
+
3220
(^d^/f
+
148890
dd'^f
+
87000
ddp
+
315
i?d?^P
+
5240
&d^
+
38950
dd^^
—
129000
dp
—
216
&d^P
+
4180
c^'d^P
+
42330
i^d'P
+
57060
l^dP
. —
5
\ <^eV
+
12600
i^d^^P
• —
181980
d'd^^f
• • •
ddep
—
30
! c'dJ'P
+
1275
c'dVf
• • «
c»c^«*
—
5250
ddp
—
705
i?d^^P
+
17670
c*d'd'
—
220125
dd^ef
—
46050
cd^P
+
260
c'd^ep
—
36510
&d^eP
—
63960
&d^i?
+
122800
ed^dp
— .
265
&d}^f
• • •
&d^^f
+
181600
e'dy'f
+
10595
ed^P
—
990
c^d^
—
6075
ed^^
+
159000
c^d^^
—
88125
c^P
+
1620
cd^eP
+
1820
f?d^P
+
43605
i^d'^e
+
27300
d^P
._
555
cd^^p
—
38820
i?d^^f
—
62025
c«cP
—
3375
d^P
+
1620
cd^^f
+
66650
c'd'e'
—
92500
b^^P
• • *
d'dP
—
1215
cd^e'
—
19800
Cd^'^ef
—
20610
c^^deP
—
1080
ddf
« • •
143]
TABLES OF THE COVAEIANTS M TO W OF THE BINARY QUINTIC.
303
W, 29 A (concluded).
»<^ w
• • •
a« 6V(^/»
+
34340
d^ h^i^d^ef
—
138750
a<» 6 We» 
17875
Wtp
+
30
&d^i?P
—
153480
c^dd'
—
1250
ddf 
6600
ed^p
—
370
c»c^ey
+
220125
i?d}^f
+
31150
dd'd +
4125
edi^f'
+
1800
i?di^
• « •
d'dJ'd'
+
40000
dd^e
• • •
^t'p
+
2845
i?d?eP
+
6660
c'd}^e
—
18750
dd^
• • •
c»d»e/*
+
570
^d^f
+
18400
cd^
+
2250
h^d^P +
729
c»(^ey»
+
1640
i?d^^
—
73375
U'c^^P
—
135
d*dep 
3645
^d^P
—
4185
cd^P
+
12310
c'^'dep
—
12060
d^df +
1350
i?^f
—
4050
cd^e^f
—
44225
c^V/»
—
1980
d'^dp +
1620
cd^P
+
1110
cd'f^
+
42500
&d^P
—
69220
d^ddf +
3375
cd^^P
—
4170
d^'ef
+
4350
d'd'^P
+
89550
d^dd 
2250
cd*t^P
• • •
d»e»
—
5125
d'de'f
—
41250
d^def 
3600
cd^^f
+
6075
WeP
—
45
d'd'
+
5125
d^dd +
2125
afo"
• • •
ed^p
—
2940
ed^eP
+
240975
d^df +
800
d^eP
+
3620
ed^p
—
9960
d'd^^f
—
179500
dH^d 
500
^^P
—
9310
<?eP
—
8175
d^d^d'
+
80125
d^de
• • •
rf*cy
+
8550
(^dJ'eP
—
46160
dd^P
—
109660
dd
• ■ •
^^
—
3375
d^d^d'P
+
34300
dd^f
—
122800
Vf^dp
+
210
i^d^f
—
10350
c'rfV
+
1250
ff^p
—
615
i?^
+
7375
d'def
+
178200
f^d^ep
—
1285
&d^P
—
73828
d'd^^
• • •
c'd^P
—
11420
&d>i?P
+
306900
dd^f
—
37950
c*cy»
—
3765
&de'f
—
159000
dd^d"
—
37125
i^d'P
—
3155
i^d^d'
+
73375
d'd^^e
+
17875
&i^i?P
+
3160
i^d^ep
+
71610
d'd^
—
2125
&d^^p
+
9850
&d^^f
—
289800
¥d^ep
+
1620
(^d^f
+
19800
<^d^^
• % %
d'dp
+
30510
&^
+
3375
^d^P
—
59835
d^dd'P
—
15120
ed^tp
—
13500
(^dd^f
+
129000
d'i^f
+
6525
ed>^p
+
50550
C>(^6*
+
80500
d'^d^ep
—
128490
ed^^f
—
62100
^d^tf
+
25050
d'^d'e'f
+
69000
ed"^
• • «
c«^e»
—
80125
d^dd
—
19875
cd^P
—
7240
cd}'f
— 
8700
ddp
+
56110
c^^P
+
28710
crf^V
+
19875
dd^ey
+
88125
cd^t^f
—
38950
cf'e
—
1125
ddd
—
40000
cd^^
+
25875
h^dp
+
990
dd^ef
—
103950
d^eP
—
6155
i^^P
+
1710
ddd
+
37125
(fey
+
12500
^d'ep
+
34620
dd^f
+
22275
d*^
—
7375
f»d^P
+
4650
ddd
« • •
h^ep
—
45
i?^f
+
7050
dd?t
—
4125
d^dep
+
615
ed'P
+
92290
ddP'
+
500
c»«y»
+
3880
ed^^p
—
243000
h'd^dP
—
7290
&d^P
+
4300
ed^^f
+
92500
d^dp
+
810
&d^^P
+
13430
edd^
—
42500
d^d'eP
+
34155
&d^P
+
18750
<^(^ep
—
219730
d^ddf
—
15300
&^f
—
2175
i*d^^f
+
318500
d^d
+
1125
i^d^eP
+
30040
d'd^
—
80500
d'd'P
—
14895
^d^e'P
—
101450
&d:^p
+
114960
d'd^df
—
27300
t^d^^f
—
1850
&^^f
+
5250
d^d^d
+
18750
d'dfP
—
25875
c»cfe*
« • •
d'^dtf
+
30250
304 TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. [l43
For the lower covariants the numerical verifications are given for the entire
coeflScient, but for the higher ones where the number of terms in a coeflScient is con
siderable they are given separately for the diflferent powers of a; and it is also
interesting to consider them for the separate combinations of a and b. I recall that the
positive and negative numerical coefficients are summed separately, so that (+a number)
means that the sum of the positive numerical coefficients is equal to the sum of the
negative numerical coefficients and thus that the whole sum is =0.
It is to be observed that for the lower covariants the sums of the numerical
coefficients do not vanish for the separate powers of a: thus in the invariant 0, 141,
the sums of the numerical coefficients for the terms in a\ a^ a® are =1, — 2, 1
respectively.
As regards the invariants Q and Q\ for the first of these, Q, the sums of the
numerical coefficients for the terms in a*, a', a', a\ aP are each of them =0, but this
is not the case as regards Q' ; in fact Q' is = G' 4 a multiple of Q ; hence the sums
for Q are the same as those for G*, viz. they are =1, —4, +6, —4, +1 respectively.
Like results present themselves in other cases, and they might probably be accounted
for in a similar manner; we have a series of sums not each =0, but which are equal
to a set of binomial coefficients taken with the signs + and — alternately and thus
the sum of these sums is = 0.
For R, 8 and 8\ I have given the sums for the different powers of a; and
in regard to iS I give here the following paragraphs from the Tenth Memoir on
Quantics : —
I remark that I calculated the first two coefficients So, Si, and deduced the other
two, Sa from 8i, and S, from 8o, by reversing the order of the letters (or which is
the same thing, interchanging a and /, b and e, .c and d) and reversing also the signs
of the numerical coefficients. This process for flfj, flf, is to a very great extent a veri
fication of the values of So, 8i. For, as presently mentioned, the terms of So form
subdivisions such that in each subdivision the sum of the numerical coefficients is
= : in passing by the reversal process to the value of flf,, the terms are distributed
into an entirely new set of subdivisions, and then in each of these subdivisions the
sum of the numerical coefficients is found to be = 0; and the like as regards Si and S^.
If in the expressions for So, S^ 8^, S, we first write d = e=f=l, thus in eflfect
combining the numerical coefficients for the terms which contain the same powers in
a, 6, c, we find
So = a' ( 2c» + 6c»  6c 4 2)
h a« {6»(6c«  12c  6) + 6( 15c' 4 33c» 21c 4 3)
4 b' (42c*  147c» 4 195c"  117c 4 27)}
4 a {b*. 4 6» (30c*  36c 4 6) 4 6' ( llTc* 4 249c«  183c 4 51)
4 6 (9c« 4 148c*  378c' 4 330c»  99c) 4 b' ( 63c« 4 166c*  147c* 4 45c»)}
143] TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. 305
+ a^{6^2 + 6»(15c + 3) + 6*(76c>69c + 24) + &»(9c*167c'i225c«87c2)
+ 6» (72c» + 48c*  186c» + 96c») + 6 ( 126c« + 201c»  87c*)
4 6* (27c«  4f6c' + 20c«)}
which for c = 1 becomes
= 26«  126» + 306*  406» + 306« 126 + 2, that is 2 (6  1)«,
and for 5 = 1, becomes =0.
8, = a» (Oc» + Oc + 0)
+ a« {6» (Oc 4 0) + 6 (3c^  9c» 4 9c  3) + 6* (24c*  99c» + 163c»  105c + 27))
+ a {6*.0 + 6'(6c« + 12c6) + 6''(24c« + 90c''108c + 42)
+ 6 (33c*  90c» 4 54c» 4 30c  27) 4 6* ( 27c« 4 78c«  66c* 4 6c» 4 9c»)}
+ a« {6* (3c  3) 4 6* ( 1 5c 4 15) 4 &» (6c»  1 2c'» 4 36c  30)
4 6' (9c»  42c* 4 84c»  lOBc^ + 57c) 4 6 (9c«  54c» 4 96c*  Sic*)
4 b' (9c'  9c«)}
%vhich for c = l becomes =0.
/S% = a» (Oc 4 0)
4 a» {6» . 4 6 (Oc« 4 Oc 4 0) 4 6' (18c*  72c^ 4 108c*  72c 4 18)}
4 a l&»(0c40)46»(33c» + 99c«99c433)+6(57c*162c»4144c«30c9)
4 6" ( 60c« 4 207c*  261c» 4 141c"  27c)}
4 a« {6» . 4 6* (15c«  30c 4 15) f 6» ( 54c» 4 102c«  42c  6)
4 5» (123c*  297c» f 243c«  87c 4 18) 4 6 ( 27c« + 102c*  96c» 4 21c=)
4 6" (27c'  60c« 4 51c«  12c*)}
'^'V'hich for c = l becomes =0.
^= a».0
4 a* {5 (Oc 4 0) 4 6« (Oc» 4 0c» 4 Oc 4 0)}
4 a {6».0 45»(0c"40c40)46(9c*436c'54c*436c9)
4 &• (36c»  171c* + 324c»  306c« 4 144c  27)}
4 a' {6* (Oc 4 0) 4 &» (7c»  21c« 4 21c  7) 4 6»( 39c* 4 135c' 171c» + 93c  18)
4 b (66c*  243c* 4 333c»  201c» 4 45c)
4 6^ ( 27c' 4 101c«  141c« 4 87c*  20c»)}
'^^liich for c = l becomes =0.
It follows that for c = rf = e=/=l, the value of the co variant S is =2(6 — 1)V,
"^^liich might be easily verified
C. n. 39
306 TABLES OF THE COVARIANTS M TO W OF THE BINABT QUINTIC. [143
For T, U, V and W, I look at the sums for the different combinations of a and £
Thus for T we have
a;
coefficient.
y coefficient.
a*b^
26
26
a^b'^
=
12
a»6>
*
14
M\/
a»6«
*
2
b^
141
6*
112
b^
281
436
6«
281
aH*
^
1
zUV/
a»6»
A
42
6»
106
6»
546
6«
186
b'
696
b'
1173
b'
366
b'
2272
3738
a^l^
*
16
*j t tf\j
a} I/'
*
5
6<
359
b*
179
6»
1411
b*
821
6«
3103
6'
2097
b^
3030
b"
2147
b^
1197
9116
b'
1262
a«6'
«^
2
«7 X X V
6«
92
—
78
an«
A
28
6»
307
—
349
6»
342
6^
1073
—
1003
b*
1790
6»
2040
—
2110
6»
3496
6«
1930
—
1880
6«
3445
6>
1207
—
1221
6>
2064
6«
231
—
239
6880
6<>
463
i
12
395
1650
6511
11628
A 20196 ^ 20196
Observe here that in the ^^coefficient for the terms in a® the successive sums
are 2, 4 14, 42, +70, 70, +42, 14 + 2, which are the coefficients of 2(tfiy.
TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 307
^r U we have
^ 36
^ 24
b'
198
b'
242
^ 2
6'
208
b'
286
b'
866
6^
1246
^ 64
b*
328
6»
1258
b^
2586
b'
2186
b'
856
dc 4
b'
70
6»
448
b*
1488
6»
2140
6=
1678
b'
884
b'
166
36
464
2608
7278
6878
17264
39—2
308
TABLES OF THE COVABTANTS M TO W OF THE BINARY QUINTIC.
[143
For V we have
X coefficient.
a»6«
^ 36
a*b*
^ 20
b'
284
b'
1094
a'b*
2
6»
184
6»
1656
b'
3624
b'
4898
an»
* 14
6*
666
6»
6608
6«
10512
b^
22042
b^
9162
a^b""
4
6«
76
48
6»
2956
 3040
6*
11946
^ 11806
6»
23924
 24064
6«
25110
 25026
b'
25524
 25552
6«
8822
 8812
a«6«
18
6^
184
324
6«
4098
 3622
6»
19350
 20274
6*
42398
 41278
6»
51872
 52740
6«
44320
 43900
6^
20624
 20740
b^
3870
 3856
36
1398
10364
49004
=t 98358
db 186734
345894
a»6«
b""
y coefficient.
^ 24
4
144
436
a»6»
^
24
6»
776
¥
2696
¥
1264
a»6»
^
6
b*
300
6»
2236
6«
8616
¥
15442
¥
33044
a}b'
^
78
6»
852
b*
8310
b^
30200
6»
56740
b'
39956
b'
17986
aPb^
2
V
286
—
270
6«
2026
—
2082
6»
9360
—
9248
¥
19760
—
19900
b^
36442
—
36330
b*
30340
—
30396
b'
23426
—
23410
b'
5120
—
5122
24
584
4760
59644
154122
126760
345894
Here in the ^^coefficient for a^ the successive sums are — 4, +28, — 84, 4 140,
— 140, + 84, — 28, + 4, which are the coefficients of — 4 (tf — 1)' ; and for a® the successive
sums are 18, 140, +476, 924, +1120, 868, +420, 116, +14, which are the
coefficients of 18 (tf — 1)^ + 4 (tf — 1)'. In the ycoefficient the successive sums are
2, +16, 56, +112, 140, +112, 66, +16, 2, which are the coefficients of
2(tfl)».
] TABLES OF THE C0VARIANT8 M TO W OF THE BINARY QUINTIC.
309
Finally for W we have
2972759
7A0
a' 6
16
aH
fc 175
h^
806
a^h"
* 80
6«
1175
6»
2760
¥
6871
a*b*
Ik 570
b^
5200
6«
18005
6*
44720
b'
23810
a*6*
't^ 90
6»
2386
b*
26675
6*
84680
6«
107730
b"
199160
b'
240499
a«6'
* 15
6^
640
&•
8260
6»
59135
6<
182055
6«
341470
6»
699260
6»
612015
b^
304501
16
981
10886
92305
661220
a^6»«
+ 1
6^
120 
130
6»
1125 
1080
6'
30350 
30470
6«
122400 
122190
6»
332494 
332746
6*
729150 
728940
6»
880750 
880870
6«
466935 
466890
6^
363670 
363680
60
76116 
76115
a«6"
+ 
5
b'^
400 
346
6»
3500 
3765
6»
26240 
25460
6^
154030 
155560
6«
409700 
407600
6»
747985 
750043
6*
745920 
744480
6»
613100 
613805
6»
311790 
311560
6^
89215 
89260
bP
9999 
9995
3003111
3111879
9087749
2207351
Here for the terms in a* the successive sums are
1^ _10, +46, 120, +210, 252, +210, 120, +45, 10, +1,
;h are the coefficients of (tf ly^ and for the terms in a* the successive sums are
«5, +54, .265, +780, 1530, +2100, 2058, +1440, 705, +230, 45, +4,
± are the coefficients of  5 (tf  1)"  («  ly.
310
[144
144.
A THIRD MEMOIR UPON QUANTICS.
[From the PhiloaophiccU Tramactiona of the Royal Society of London, voL cxlvi. for the
year 1866, pp. 627—647. Received March 13,— Read April 10, 1856.]
Mt object in the present memoir is chiefly to collect together and put upon
record various results useful in the theories of the particular quantics to which they
relate. The tables at the commencement relate to binary quantics, and are a direct
sequel to the tables in my Second Memoir upon Quantics, voL CXLVI. (1856), [141].
The definitions and explanations in the next part of the present memoir are given
here for the sake of convenience, the further development of the subjects to which
they relate being reserved for another occasion. The remainder of the memoir consists
of tables and explanations relating to ternary quadrics and cubica
Covariant and other Tables, Nos. 27 to 50 (Nos. 1 to 50 binary quantics)^
Nos. 27 to 29 are a continuation of the tables relating to the quintic
(a, 6, c, d, e, fjx, yf.
No. 27 gives the values of the different determinants of the matrix
( a, 46, 6c, 4d, e )
a, 46, 6c, 4d, e
6, 4c, 6d, 4«, /
6, 4c, 6d, 4«, /
determinants which are represented by 1234, 1235, &c., where the numbers refer to
1 The Tables 49 and 60 were inserted October 6, 1S56.— A. C.
144]
A THIBD MEMOIR UPON QUANTICS.
311
the different columns of the matrix. No. 28 gives the values of certain linear
functions of these determinants, viz.
2,=
1256 +
2345
 2 . 1346,
L'^i
. 1256 
1346,
8M = 
• 1345 + 2
.1246,
8ir=
2346 + 2 .
, 1356,
8iV=
' 1245 + s
. 1236,
8N' = 
2356 + 8 .
. 1456,
80P =
L'3L
= 8.
1346  8
16P' = 
5L' L
= — 18.
1256  8
^t the end of the two tables there are given certain relations which exist between
tr^he terms of Tables 14, 16, 25, 26, 27 and 28.
No. 27.
1234.
12d6.
1236.
1246.
1246.
1345.
1256.
2345.
«V
aV ^
a*d/+ 6
a'd/ 6
a V + 4
a'c/"
ay«+ 1
ay«
«»«  16
a^de + 24
aV
aV + 16
oi^ 4
a6^ 24
o^ 2
a6«/*
«V + 36
aiy+ 4
abc/^ 22
abr/+ 6
a6e« 4
abe^ + 64
occf/" 16
acd/+ 20 i
^3i«e+ 16
abee 84
abde 6
ahde 26
o^y 24
acy+ 24
octf« + 16
ace*  80 ;
^3bcd 152
oW" 24
ac^e + 16
ac««  96
acde\ 24
acde 208
acPe + 16
ad^e+ 60
«j* + 96
a(j«rf+ 64
acd^
acd^+ 96
ocP
ew^ + 144
6V/ 15
6*^ 80
*(i + 80
b^e + 60
by + 16
6y ...
6»c/ + 24
6V
6»62 ...
6V + 240 ;
«V  60
i^cd 40
b'ce  10
b^ce + 90
bHe  20
6»c^ 40
6cy ...
be"/ + 60 ;
OCT ...
6«rf« ...
6»c?»  80
oc^c ...
6c»6 + 60
6crfc
6cde 860 1
bc^d
bc^d
6ccP
bed" 40
6rf»
6c^ + 960 ■
1
C/ • . .
(j . • •
c»c/
c'c/
c»«
c»e + 960 j
i
1
c'd' ...
c'd'  320 j
1346.
2346.
1356.
2356.
1456.
2456.
3456.
ay* ...
ai^
a^+ 4
acP 6
axp + 6
a<^« 4
a^"
a6c/'+ 16
ac«/* 24
cLcef — 4
adtfv 6
adef 22
acy+ 4
6(^ 16
acc(/' 36
a(^/+ 24
a(/y 24 ae*
a<j» + 16
6c/« + 24
6«y + 16
ac««  16
a{2s*
a^ + 24 ^»y« + 16
6y* ...
bdef^ 84
cy« + 36
ad^e + 36
6««/+ 64
V'tf 4
6cc/ 26
bcef^ 6
ftc» + 60
cdef 152
l^df 16
bcdf^ 208
6<j(^+ 24
6«((/ 96
6c^/+ 16
c»c/  24
ce» + 80
6V ...
6<j««  40
6ce«  20
6de» + 90
6<^ 10
ccZy+ 64
c^/ + 96
6cy+ 36
6rf«e + 60
bd^e
i^df + 96
d'df ...
cc^  40
d?^  60 !
bcde 20
c»/ + 144
cy ...
e^  80
vTVi ...
d^t
1
W»
c»cfe 40
c*cfe
ccPtf
cc^e
V c ...
CflP
CflP
cf*
d^
<?d^
1
312
A THIRD MEMOIR UPON QUANTICS.
[144
No. 28.
N.
M.
L.
L'.
P.
F.
M'.
N'.
aHf + 3
a V  2
ahcf 9
abde+ 1
a V + 1
abd/+ 2
a6c» 9
ocy 9
ay « + 1
a6c/ 34
aor//*+ 76
acc«  32
ay«+ 3
oic/ 22
acdf 12
occ' + 64
ay> ...
ahef+ 1
oo// 3
oc^ + 2
ay« 1
aA«/'+ 9
occ^ 1
aci^ ^ 18
abf^ 1
ac«/*+ 2
(w^/ 9
a<i«»+ 6
ac/^+ 3 ^
oic/ 9
oc* + 6 \
6y« 2
cu^e + 18
acdei 32
ad'e 12
a<Pe 36
cuPe
o(f"e+ 12
bhf 9
bee/ + 1
acd^ 12
b*/ + 6
o^  18
6 V + 6
6V/ 32
6V + 225
6V/+ 64
6V  45
b'd/+ 2
6V  9
b^df 18
6V
6«i/'+ 32
6dy+ 181
W«»  1^ ;
b^ce  15
b^d" + 10
b^de
bc'e  15
bc^f 12
6cfl?c 820
bey 36
6ccfe+ 20
bc\f ...
6<;flfe+ 31
6cy + 12
ftcSc + 45
bd'e 15
cy  18
c^ 12^ \
C*(5« + lO^ !
6c'cZ
bcd^^ 10
b<P + 480
M»
bd*  18
6ii»  30
c'de + 10
cd^e ... *
c*
C^(/
c^e + 480
CJ C/ • • •
c«e  18
c'e  30
ci*
c^ ... '
c'd'  320
c"flP
c'd' + 12
c«cZ« + 20
If the coeflBcients of the table 14 are represented by ^A, B, ^C, viz. ¥rritiDg
il = 2 (oc  46d + 3c»),
B = a/ 36e + 2cd,
(7 = 2(6/4ce+3d«),
then we have the following relations between 1234, &c. and A, B, C, viz.
1
+ Bx
+ Ax
1234 =
+ 6a«
12a6
+ 16 oc 10 6»
1235 =
+ 6 ah
 2 oc  10 6>
+ 6ad
1236 =
 2ac+ 8 6=
+ 6<k£ 18 6c
 2d/+ 8c»
1245 =
+ 18ac
 6 CM?  30 6c
+ Sae +10 bd
1246 =
+ 12 6c
+ 4a<5 4 6rf24c«
+ 46c + Scd
1345 =
+ 24 ac£
 8 oc  40 6rf
+ 4 a/ + 20 6c
1256 =
 I ae + 4 6rf +
3c»
+ I a/ + 5 be  IS cd
 lb/ + 4cc+ 3dr
2345 =
+ 20 oc + 40 6c/ 
30 c«
 SO be + 20 cd
+ 20 6/ +40 cc 30 (?
1346 =
+ 4 06 + S bd +
6c»
36 erf
+ 4 6/+ 8cc+ 6d^
2346 =
+ 4 a/ + 20 6c
 8 6/ 4cc
+ 24c/
1356 =
+ 4 6c + 8 cc?
+ 4 6/ 4c«24rf«
+ 12cfe
' 2356 =
+ 8 6/ + 10 cc
 6c/30cfo
+ lSd/
1456 =
+ 6cc
+ 6 c/ ISde
 2c(^+ 8c»
2456 =
+ 6 c/
 2((/*10c»
+ 6 c/
3456 =
+ 16 df 10 e^
12 c/
+ 6/«
and the following relations between L, L\ &c. and A, B, C, viz.
Cx
+ Bx
+ Ax
^=
 3 oc + 3 6«
+ 3 orf  3 6c
 lac+ 1 6rf
3/ =
 Sad+ 3 6c
+ 3ac  3c'
 l€/+ led
Z =
+ 11 oc + 28 6rf39c»
+ 1 a/  75 6c + 74 cd
+ 11 6/+ 28 cc 39 rf«
Z'
 7ac+ 4 6rf+ 3c'
+ 3 a/ + 15 6c  18 erf
 76/+ 4cc+ 3rf»
2iP =
 lac 2bd+ 3c»
+ 3 6c  3 erf
+ 1 h/+ 2 ec Srf"
7^ =
+ 3ac 6bd+ 3c«
 1 a/+ 1 erf
+ 3 6/ 6 oc + 3 rf» J
if'=
 I a/ + I cd
+ 3 6/ 3rf«
 Sc/+ 3rfc I
jr=
 16/+ 1 cc
+ 3c/  3rfc
 3(i^+ 3c» 1
_ iiiJiMOIIl UPON QUANTICS.
313
» e nave also the following relations between i, L\ &c. and a, 6, c, d, c, /, viz.
aP 6Jf 4 ci^T =0,
aN'+2bM''cr
+ 3eN
= 0,
= 0,
3cJ\r'2dif'4 eF+fM=0,
The quartin variant No. 19 [G] is equal to
i.e. it is in fact equal to —4 into the discriminant of the quintic No. 14, [A].
The octin variant No. 25 [Q] is expressible in terms of the coefficients of Nos. 14
and 16, viz. A, B, C, as before, and Ja, /8, 7, JS the coefficients of No. 16, [D], i.e.
a = 3 (ace ad^ h^e +26c(Z c»),
/8= acfade l^f+ bd^ + bee  d^d,
y = adf—ae^ — bcf+ bde + d^e —cd^,
S = 3 ibdf" 6e» 4 2cde  c*/  d»),
then No. 25 is equal to
A, B, C
a, ^, 7
The value of the discriminant No. 26, [Q'], is
(No. 19)«128 No. 25. [that is Q' = G«  128Q.]
We have also an expression for the discriminant in terms of Z, L\ &c., viz. three
times the discriminant No. 26 is equal to
[or say 3Q' =] LU 4 UMM'  &4^NN\
remarkable formula, the discovery of which is due to Mr Salmon.
It may be noticed, that in the particular case in which the quintic has two square
^tors, if we write
(a, 6, c, d, e, /$a?, y/ = 5 {(p, 5, rjx, y)«}« . (\, /A$a?, y).
C. II.
40
314
A THIBD MEMOIR UPON QUANTIC8.
[144
then
a = 5\p^, h = 4tpqK 4 J5 V,
and these values give
P =Z(63»pr),
M = K, lOpq,
where the value of iT is
c = (23" +pr)\ + 2pqfi,
d = 25r\ + {2f +pr)fi\
8 (pfi^  2qfi\ 4 rVy (/)r  j«)*.
The table No. 29 is the invariant of the twelfth degree of the quintic, given i^
its simplest form, Le. in a form not containing any power higher than the fourth o^
the leading coefficient a: this invariant was first calculate by M. Faa de Bruno.
No. 29. [See U. No. 29, p. 294]
The tables Nos. 30 to 35 relate to a sextic. No. 30 is the sextic itself ^
No. 31 the quadnn variant ; Nos. 32 and 33 the quadricovariauts (the latter of them,
the Hessian); No. 34 is the quartinvariant or catalecticant ; and No. 35 is the?
sextinvariant in its best form, i.e. a form not containing any power higher than the
second of the leading coefficient a.
No. 30.
a^\
6 + 6
c + 15
d+20
e + 15
/+6
g+l Wx, y)*
No. 31.
No. 32.
ag
+
1
¥
—
6
ce
+
15
d^
—
10
^16
a« + 1
W  4
c« + 3
a/ + 2
6tf  6
cd + 4
0^+1
C6  9
(^ + 8
hg ^ 2
cf  Q
de + 4:
eg + I
^/ 4
e» + 3
:fe4
±G
No. 33.
*9
^6
5«, 2/y
ac + I
ad+i
otf + 6
a/+ 4
ag+ 1
6^+4
eg + ^
dg + 4:
eg + 1
("'
6c 4
bd+ 4
6<»+16
6/+ 14
C/ + 16
d/+ 4
e/4.
rl
c« 10
cd 20
c« + 5
cfo  20 i c«  10
1
<^20
i
i i
$*. y)'
^1
^4
:l=10
db20
db20
Jk20
A 10
^4
:fcl
144]
A THIRD MEMOIR UPON QUANTICS.
315
No. Si.
No. 35.
aeeg + 1
acP 1
ad^g  1
adef + 2
o^ 1
h'eg 1
h'P +1
bcdg + 2
6ct/^ 2
6dy2
6d;e» +2
1
+ 2
+ 1
3
+ 1
12
a^<P(^ + 1
a^defg— 6
aV/» + 4
aV<7 + 4
aV/>  3
abcd^— 6
a^«/gr+ 18
a^/» 12
aMy^+ 12
ahde^gl^
aMf + 6
ao*5^ + 4
ac»c«^  24
o^dfg\%
a^eP + 30
ac<Peg + 54
acdy^12
acrfey42
ace^ +12
orf*^ 20
««/'«/• +24
acP^  8
W^ + 4
Wg 12
6y» + 8
6V^  3
ftW^r +30
6»c«/>  24
b'iPeg  12
6«rfy*  24
6»dey + 60
6V  27
6cy^ + 6
hi^deg  42
bccPg
bccPef
bcd^
bdy
d^eg
c'def
ed^e
+ 60
30
+ 24
84
+ 66
+ 24
24
+ 12
27
 8
+ 66
 8
24
39
+ 36
 8
:A:665
The seztmvariant may be thus represented by means of a determinant of the
dxth order and of the quadrinvariant and quartinvariant.
5xNo. 35 =
44(asr66/+15ce
a, 26, 3c,
4d,
e
6, 2c, 3d,
4c,
f
c, 2d, 36,
4/
9
a, 46, 3c, 2d,
c
6, 4c, 3d, 2c,
/
c, 4d, 3c, 2/,
9
10*)
a, 6, c.
d
6, c, d,
e
c, d, c,
f
d, c, /,
9
The tables Nos. 36 and 37 relate to a septimia No. 36 is the septimic itself;
^^0. 37 the quartinvariant.
No. 86.
( o+l
6 + 7
c + 21
rf+35
« + 35
/+21
9 + f
h+1
40—2
316
A THIRD MEMOIB UPON QUAMTICS.
[U4
No. 37.
aW  1
6rf«A
 40
abgh ' + 14
bdeg
 50
acjh  18
bdp
 360
acg*  24
bf?f
+ 240
ad^h + 10
<?eg
 360
ad/g + 60
c>P
 81
ae^g  40
cd^g
+ 240
b^h  24
cdef
+ 990
6y  25
ce"
 600
bcfg + 234
ciy
 600
bceh + 60
(£«c»
+ 375
±2223
The tables Nos. 38 to 45 relate to the octavic. No. 38 is the octavic itse
No. 39 the quadrin variant ; Nos. 40, 41 and 42 are the quadricovariants, the last
them being the Hessian; No. 43 is the cubinvariant ; No. 44 the quartinvariaut, ai
No. 45 the quintinvariant, which is also the catalecticant.
No. 38.
(
a+ 1
6 + 8
c + 28
(f + 56
« + 70
/+56
^ + 28
A + 8
» + l W^^yf
No. 39.
No. 40.
ai
+
1
bh
—
8
eg
+
28
df
—
56
c«
+
35
±64
ag + \
oA + 2
ai + 1
bi + 2
ci + 1
6/ 6
bg  10
bh  2
ch  10
dh Q
( cc + 15
cf + 18
eg  S
dg + IS
eg + lb
d«  10
de  10
d/ + 34:
e/  10 /«  10
e«  25
±16
±20
No. 41.
±35
±20
±16
5^, y)^
oe + 1
a/+ 4
ag + Q
ah + i
ax ^^ 1 6t + 4
ci + 6
c^i + 4
et + 1
6c£  4
be 12
bf  S
bg + S
bh + 12 cA + 8
c^A 8
eA12
/A 4
c« + 3
cd + S.
ce 22
cf 48
eg  22 ety  48
c^  22
fg^ 8
/ + 3
cf» + 24
cfe + 36
df  36 «/ + 36
/' + 24
6^+45
±4
±12
±30
±48
±58
±48
±30
±12
$a, yf
No. 42.
ac + 1
ad^r 6
ac+ 15
a/'+20
a^+ 15
aA+ 6
1
ai + 1 6i + 6 ci + 15
rfi+20
«» + 15
/i + 6'j,i +
6'l
6c 6
6c£+ 6
6c +50
6/+ 90
bg^ 78
6A+ 34
cA+ 78
dh^ 90
cA + 50
/A+ 6
yA6 A»
/
c« 21
ccf.70
ci« 105
c/ + 126
c^ + 154
cfy+126 />105
/^70
^•21
(
1
1
(fo210
rf/ 14
6» 175
e/  210
±1
±6
±21
±70
±105
±210
±189
±210
105
±70
±21
±6
14]
A THIRD MEMOIR UPON QUANTICS.
317
No. 43.
No. 44.
aei + I
a/h 4
a^ + 3
bdi  4:
heh + 12
hfy  8
c»i + 3
cdh  %
ceg  22
cP 4 24
J^g + 24
(ii'/  36
e* +15
±82
ac<7t  1
hcgh + 3
cd^i  2
acA« + 1
hdei + 1
cdeh  23
adfi + 3
6c(A  10
c^//*y + 27
ac&/A  3
hag"" + 9
ce»y + 19
oeH  2
6eVi + 11
cef  21
a€/7t + 1
hefg  23
(^A + 12
alf + 3
hp + 12
(r€g  21
oPg  2
c'et + 3
c^/'  13
6Vi + 1
<?fh + 9
de"/ + 32
6^/i*  1
c«/  12
e*  10
6r/i  3
1
±147
No. 45.
1
i
acegi
+
1
«/*
+
1
bde(f^
4
CiPg^
+ 1
t
aceJi^
—
1 I
b^egi
1
hiiPg
+ 2
cdefg
 2
acfi
—
1
IPeh'
+
1
be>h
2
cdp
2
acfgh
+
2
I'M
—
2
beVg
V 4
c^g
3
acg^

1 !
byi^
+
1
hop
 2
ce^dh
^ 4
(vPql
—
1
by
+
1
egi
 1
ccT
^ 3
1
(uPh^
+
1
bcdgi
+
2
eh^
+ 1
dH
t 1
1
1
adefi
+
2
bcdli"
—
2
<?dfi
+ 2
d'eh
2
Ofiegh
—
2
bcefi
—
2
<?d<jh
2
<^fy
2
a<//Vi
—
2 1
bceffh
+
2
eeH
+ 1
d'e'g
h 3
1
ad/g'
u
2 1
br/Vi
+
2
e'efh
 4
it'^P
+ 3
1
a^i
—
1
kf</'
—
2
oV
+ 2
: d^f
 4
1
1
ae^jh
T
o 1
bdifi
—
2
cyv
1 1
^
 1
a^g^
+
1 '
bd'gh
+
2
cd^ei
 3
1
aepg
3 1
bd(^i
+
cd^fh
+ 2
±56
If we
write
i' ■
d
612\,
No. 39 = /,
No. 43== J,
No. 44 = ir,
No. 45 = £,
lambdaic, viz.
h . c , d ,
e12X '
c , d , C + 3X,
/
d , e2\, f
9
e+3\, / . g
I
f ,9 , h
•
\
318
is equal to
A THIRD MEMOIR UPON QUANTICS.
[144
L + 2\K + 3X»J+ 18V/  2592V.
Nos. 46 to 48 relate to the nonic. No. 46 is the nonic itself; Nos. 47 and 48
are the two quartinvariants, each of them in its best form, viz. No. 48 does not
contain a', and No. 47 does not contain aci^, the leading term of No. 48. The
nonic is the lowest quantic with two quartinvariants.
No. 46.
a+l
6 + 9
c + 36
c^ + 84
e + 126
/+126
^ + 84
/i + 36
1+9
i+1
No.
47.
No. 48.
a'/  1
b/^h
a«/ ...
bph + 70
abii + 18
b/g^  720
dbij
bfg^  45
act*
cYj + 432
acv" + 2
c^^gf + 27
achj  72
<?gi  1728
achj — 2
i^gi  52
\ adgj + 168
c W
adgj + 7
<^h^ + 25
i adhi
cdej  720
orf/a 7
ccfe; — 45
a^fj  108
cdfi + 2160
a«^  5
cdf% + 23
aegi  576
cdgh+ 4608
ocgri 22
ce^A + 22
aeh^ + 432
cfh,
aefi^ +27
c(H + 70
afH + 540
cefh  2592
a/«t +25
cefh 127
afgh 720
eeg"  5760
afgh 45
c«^ + 32
a^ + 320
c/V + 4320
a^ +20
cpg + 25
6«A;
d?j + 320
6*/*/ + 2
cO' + 20
W  81
d^'ei  720
W  2
d^ei  45
hcgj
d'fh 5760
6c5[/  7
cP/A+ 32
hchi + 648
d«5^  1536
tcAt + 7
d'g^ + 47
hdjj  576
dc/^ + 14688
bdjj  22
<*!/& + 85
bdgi + 792
de^h + 4320
W^t + 74
d^h + 25
bdh^  1728
dp  8640
bd^i^  52
df*  50
begh + 2160
e»<7  8640
ft^^r/* + 23
e*^  50
b^j + 540
cy« + 5184
be'j +25
ey« + 30
be/t  972
6(2/t 73
3[a^ y)*
:k 41650
:A=698
Nos 49, [49 a] and 50 relate to the dodecadic. No. 49 is the dodecadic itself:
[No. 49 A, inserted in this place, but originally printed in the Fifth Memoir on Quantics,
is the dodecadic quadricovariant]. No. 50 is the cubinvariant. [The numerical coefficients
in this last table as originally printed in the Third Memoir were altogether erroneous,
and the table as here printed is in fact the table No. 60 j^, of the Fifth Memoir on
Quantics.]
No. 49.
a+l
6 + 12
c + 66
rf+220
c + 495
/+792
^ + 924
A + 792
i+495
J +220
A;+66
f+12
m + l
5«,yr
144]
A THIRD MEMOIR UPON QU ANTICS.
319
16
No. 49 A.
ag^ 1
ah+ 6
at + 15
a;+ 20
aki 15
6/ 6
6<730
bh 54
bi 30
6; + 30
c« + 15
C/ + 54
eg + 24
cA  150
ci 270
( d'lO
cfo30
rf/+150
dg + 430
^+270
e«  135
./  270
c^ +495
/2  540
aZ + 6
bk+ 54
c;  150
di 270
«A + 1080
fg 720
am + 1
bl + 30
cA; + 24
dj 430
c» +495
//i +720
^ 840
=1:60
:l=189
:t460
rtSlO
:tll40
±1270
brn + 6
c/ + 54
dk  150
ei  270
/» +1080
gh  720
cm + 15
dl + 30
cifc 270
^ +270
^i +495
h^ 540
c/m + 20
el  30
/>fc 150
<d +430
Xi 270
cm + 15
/I  54
i^ifc + 24
hj +150
135
t«
/m + 6
^/ 30
A>fe +54
y 30
gm + I \
A/  6 '
tA; + 15 ty \n
j2 ^10 i$*»y)"
±1140
±810
±450
±189
±60
±16
No. 50.
agrn + 1
c/l  54
dhi + 270
oA/  6
c^A; + 24
e^A; 135
aik + 15
chj +150
^Jj +270
a/  10
ci«  135
c^i + 495
bfm — 6
<Pm  10
eh* 540
6^; + 30
rfe^ + 30
/S  540
bhk 54
dfk +150
fgh + 720
6v + 30
^^ _430
^ 280
cem + 15
±2200
Resuming now the general subject, —
54. The simplest covariant of a system of quantics of the form
(where the number of quantics is equal to the number of the facients of each
quantic) is the functional determinant or Jacobian, viz. the determinant formed with
the differential coeflScients or derived functions of the quantics with respect to the
several facients.
65. In the particular case in which the quantics are the differential coefficient8 or
derived functions of a single quantic, we have a corresponding covariant of the single
quantic, which covsuriant is termed the Hessian ; in other words, the Hessian is the
determinant formed vrilth the second differential coeflScients or derived functions of the
quantic with respect to the several facients.
66. The expression, an adjoint linear form, is used to denote a linear function
fe + i7y+ .... or in the notation of quantics (f, i;,...$a?, y,...), having the same facients as
320 A THIRD MEMOIR UPON QUANTICS. [144
the quantic or quantics to which it belongs, and with indeterminate coefficients
(f, i;,...) The invariants of a quantic or quantics, and of an adjoint linear form, may
be considered as quantics having (f, 17,...) for facients, and of which the coefficients
are of course functions of the coefficients of the given quantic or quantics. An invariant
of the class in question is termed a contravariant of the quantic or quantics. The
idea of a contravariant is due to Mr Sylvester.
In the theory of binary quantics, it is hardly necessary to consider the eontra
variants; for any contravariant is at once turned into an invariant by writing (y, — d?)
for (f, 17).
57. If we imagine, as before, a system of quantics of the form
(•$^, y, ...)'«,
where the number of quantics is equal to the number of the facients in each quantic,
the function of the coefficients, which, equated to zero, expresses the result of the
elimination of the facients from the equations obtained by putting each of the quantics
equal to zero, is said to be the Resultant of the system of quantics. The resultant
is an invariant of the system of quantics.
And in the particular case in which the quantics are the differential coefficients,
or derived functions of a single quantic with respect to the several facients, the
resultant in question is termed the Discriminant of the single quantic; the discriminant
is of course an invariant of the single quantic.
58. Imagine two quantics, and form the equations which express that the differen
tial coefficients, or derived functions of the one quantic with respect to the several
facients, are proportional to those of the other quantic. Join to these the equations
obtained by equating each of the quantics to zero; we have a system of equations,
one of which is contained in the others, and from which therefore the facients may
be eliminated. The function which, equated to zero, expresses the result of the
elimination is an invariant which (from its geometrical signification) might be termed
the Tactinvariant of the two quantics, but I do not at present propose to consider
this invariant except in the particular case where the system consists of a given
quantic and of an adjoint linear form. In this case the tactinvariant is a contravariant
of the given quantic, viz. the contravariant termed the Reciprocant.
59. Consider now a quantic
(•Ja:, y,...)*",
and let the facients x, y, ... be replaced by Xx + fiX, Xyh/^F, ... the resulting function
may, it is clear, be considered as a quantic with the facients (\ fi) and of the form
144] A THIRD MEMOIR UPON QUANTICfS. 321
The coefficients of this quantic are termed Emanants, viz., excluding the first coefficient,
which is the quantic itself (but which might be termed the 0th emanant), the
other coefficients are the first, second, and last or ultimate emanants. The ultimate
emanant is, it is clear, nothing else than the quantic itself, with (X, F, ...) instead of
(x, y, ...) for facients : the penultimate emanant is, in like manner, obtained from the
first emanant by interchanging (ar, y, ...) with (X, F, ...), and similarly for the other
emanants. The facients (X, F, ...) may be termed the facients of emanation, or simply
the new founents. The theory of emanation might be presented in a more general
form by employing two or more sets of emanating fii<;ients; we might, for example,
write \x\fi,X'{vX\ \y •{■ fiY + vY\ . , , for x, y, ..., but it is not necessary to dwell
upon this at present.
The invariants, in respect to the new facients, of any emanant or emanants of a
quantic (i.e. the invariants of the emanant or emanants, considered as a function or
functions of the new facients), are, it is easy to see, covariants of the original quantic,
and it is in many cases convenient to define a covariant in this manner; thus the
Hessian is the discriminant of the second or quadric emanant of the quantic.
60. If we consider a quantic
(a, 6,...$a?, y,...)'^,
and an adjoint linear form, the operative quantic
(which is, so to speak, a coutravariant operator) is termed the Evector. The proper
ties of the evector have been considered in the introductory memoir, and it has been
in effect shown that the evector operating upon an invariant, or more generally upon
a contravariant, gives rise to a coutravariant. Any such coutravariant, or rather such
contravariant considered as so generated, is termed an Evectant In the case of a
binary quantic,
(a , b ,...$ar, y)"»,
the covariant operator
{da, 96,...$y,^)"*
may, if not with perfect accuracy, yet without risk of ambiguity, be termed the Evector,
and a covariant obtained by operating with it upon an invariant or covariant, or
rather such covariant considered as so generated, may in like manner be termed an
Mvectant,
61. Imagine two or more quantics of the same order,
(a, 6, ...][a?, yf,
(a, /8,...$a?, yf,
we may have covariants such that for the coefficients of each pair of quantics the
covariant is reduced to zero by the operators
ad^ + Idp 4 . . . ,
ad a + I3di, h . . . .
C. II. 41
322 A THIRD MEMOIR UPON QUANTICS. [l44
Such covariants are called CtmbinanUy and they possess the property of being inva
riantive, quoad the system, i. e. the covariant remains unaltered to a factor priSy when
each quantic is replaced by a linear function of all the quantics. This extremely
important theory is due to Mr Sylvester.
Proceeding now to the theory of ternary quadrics and cubics, —
First for a ternary quadric, we have the following tables: —
Covariant and other Tables, Nos. 51 to 56 (a ternary quadric).
No. 51.
The quadric is represented by
which means
cwj» 4 6y' + c^ + %fyz + igzx 4 2Jucy .
No. 52.
The first derived functions (omitting the factor 2) are —
(a, A, gjx, y, t),
(A, 6, fjx, y, z\
No. 53.
The operators which reduce a covariant to zero are
( h, 6, 2/ $3^, df, dc)'zdy,
( a, 2A, g^Ky 9b, d/)ydg,
( g, 2/ c$3a, 3*, 3/)y3„
( a, A, 2g'$dg, 3/, 3c) ^3,,
(2A, b, f^da, 3a, 3^)a3y.
No. 54.
The evector is
(3a. 3ft, 3c, 3/. dgy dk\^, 17, f)'.
144]
A THIRD MEMOIR UPON QUANTICS.
323
The discriminant is
which is equal to
No. 55.
a, h, g
h, b, f
abo  af*  b^  ch* + 2fgh.
No. 56.
The reciprocant is
f, a, A, g
Vf K b, f
which is equal to
(6c/», cag\ ahh^ gh^af, hf^bg, fg  ch\l ri, ^.
The discriminant is, it will be noticed, the same function as the Hessian. The reci
procant is the evectant of the discriminant. The covariants are the quadric itself and
the discriminant; the reciprocant is the only contravariant.
Next, for a ternary cubic, we have the following Tables :
Covariant and other Tables, Nos. 57 to 70 (a ternary cubic).
No. 57.
The cubic \a U^
(a, 6, c, /, g, A, i, j, k, Z$a?, y, zf,
which means —
aa^ + bf¥c2^ + Sfy^z + Sgz^x + Sha^ 4 %&• + Bjza^ + 3fcry" + 6lxyz,
No. 58.
The first derived functions (omitting the factor S) are
(a, k, g, I, j, hjx, y, z)\
{K b, %, f, I, kjx, y, zf,
(jy /, c , i, g, I '$x, y, z)\
41—2
324
A THIRD MEMOIR UPON QUANTICS.
[144
The second derived functions (omitting the factor 6) are
(a.
h.
j $a;,
y>
^).
(k.
b.
fl^.
y>
A
iff.
•
c l^x,
y.
A
(I.
/.
ijix,
y>
').
a
I.
gl'^'
y>
').
(h,
k.
I T§ix,
y.
')■
No. 59.
The operators which reduce a covariant to zero are
(j.
3/.
c,
2t,
9.
(o.
k.
3«7.
ii,
2j.
(3A.
b,
•
/
2«,
(A.
b,
3i.
2/.
2i.
(3j,
/.
c,
•
t,
2*7.
( a,
Sk,
S'.
21,
i.
The evector is
2Z$3a, 96, 9<, 9/, 9(,
A$9j, 9/, 9c, 9,, dg,
2kJida, 9*, 9^
ff*
9^ 9j, dk)
^][3i> ^/» 3c> 3»» ^y» 90 — '2r9y,
2Z$9a, 9*, 9^, 9f, 9,, 9*) — a:9,,
2A][9a, 9^, 9<, 9/, 9^ 9*)y9«.
No. 60.
(9a, 9^, 9c, 9/, 9y, 9a, 9<, 9;, 9*, 9f3[f, 17, (;)*.
The Hessian \a HU^
(a, A, j$ir,
(A, A;, Z $ar,
which is equal to
No. 61.
y, 2:), (A, A;, Z$ir, y, ^), (j, ^» 3^$^*
y, ^), (Ar, 6, /$a:, y, ^), (Z, / i$a:,
y, ^), (i, /, t$a?, y, 4 (jr, t, c$a?.
y.
^)
y.
^)
y.
z)
'
agkl
Mi1
cy 1
6<*l
ae/\
abg 1
bcj 1
«cA — 1
aW 1
06c — 1
aP +1
6P +1
c? +1
65^+2
ot* +1
flt/*/ +2
bf +1
«/ffl
oT + l
q/i +1
yA' +1
/»A+1
fg" +1
6»y 1
cAZ +2
oii — 1
c/l1
ail +2
6yAl
hgj +1
A/i 2
/«2
fki 2
c*» +1
# 1
bp +1
c^/ +2
eh" + 1
y/ +2
cAA + 1
j»* +1
♦*• +1
iy +1
/y +1
/»■ +1
/Hi 2
yv +1
^ +1
^k2
/yA3
•\
/y*2
g'k +1
J7W+I
j;i*2
g;A + 1
gJ^ +1
fjl +2
y%i + i
j^At2
AS +1
At» +1
hij 2
AK+1
yA/+2
fp 1
j?P 1
A^ 1
iP 1
iP 1
AP 1
Aa +2
V* 3
v.
P 2
<l(«,y,
:i=3
:i=3
dbS
=lr5
±6
:t5
±9
144]
A THIRD MEMOm UPON QUANTICS.
325
The quartinTariant w S=
No. 62.
abcl — 1
/y +1
abgi + 1
fghl + 3
acfk + 1
/a;*i
qTg^l
/Av 1
afil + 1
^? 2
aPk 1
i^ifc« +1
hch^ + 1
ghki — 1
hfh \
^A;? 2
bgjl + 1
AV +1
hij" 1
A*? 2
qf'h 1
ijkl +3
cM/ + 1
^* +1
ciife« 1
il6
The sextiu variant is 7 =
No. 63.
a^¥<^
+ 1
acpKL
24
hcfhj^
12
cfhH
 12
fgikV^
12
a^bcfi
 6
acpjk
12
hcghH
24
cfh^J^
+ 12
fh^3
12
a^hi?
+ 4
acfg]^
12
hcghjk
+ 18
cfhjJd
60
fhijl?
12
a^cP
+ 4
acfJiki
+ 18
bchHj
12
cjr^
+ 24
fim
+ 36
a^PP
 3
cicfkJ^
+ 36
hchjt'
+ 36
cgWl
+ 12
fU^
+ 24
ab^cgj
 6
acH^l
24
hcPkl
24
cgjl^
12
9"^
+ 8
ahy
+ 4
^Poi
12
hfg^hj
+ 6
chHkl
+ 12
g'hkH
12
ahc'hk
 6
afVk
+ 24
Pf9?l
+ 12
chijh^
+ 6
ipJ^P
24
ahcfgh
+ 6
apghi
+ 6
hfif
12
chkU"
12
gh^ki^
12
abcjjl
+ 12
cpgJ^
+ 12
hg'hk
12
i^iei^
+ 12
gJJdP
12
ahcgkl
+ 12
a/Hjl
+ 12
hg'hH
+ 24
fV
+ 8
gij^l
+ 36
abchil
+ 12
a/gkil
60
hfh^
+ 12
/yA«
27
gkl*
+ 24
abcijk
+ 6
afhi?l
+ 12
hfikl
+ 12
/v*
12
h^i?
+ 8
abcP
20
af?ik
+ 6
hghijl
60
Pghji
+ 36
hS*J^
24
ah/gij
+ 18
afU?
12
bgipk
+ 6
phis'
12
hit"
+ 24
ohffl
24
agJ^il'
+ 24
bgjP
12
yy^
24
hi^kl
+ 36
abg'ki
12
aht^k
12
bhPf
+ 24
ffhki
+ 36
tVifc»
27
ahghi^
12
ai»kP
+ 12
bipP
+ 12
ffi^
12
ijkP
36
abgU^
+ 36
^cP
+ 4
(?h^l^
 3
fghHl
+ 36
Z»
 8
ahi^jl
24
^9T
 3
cPh^j
+ 24
fghijk
 6
04?}^
+ 4
h(?h^
+ 4
cfgh^k
+ 6
fgh^
36
±871
The discovery of the invariants S and
ressions were first obtained by Mr Salmon.
T is due to Aronhold, the developed
No. 64.
There is an octicovariant for which we may take
eu= d^u, dyHU,
d^Hu, fa,' u, id^dyU,
dyHU, idyd^u, J 3/ u,
dHU, ifd.d^u, idyd,u.
d,HU
326
A THIRD MEMOIR UPON QUANTICS.
[144
or else
©.tr= ia, u, ^dy u, id, u
id^U, a,» HU, d,dyHU, d^,HU j
IdyU, dyd^HU, dy* HU, dJd^HU
J a, £7, d,d,HU, d.dyHU, a,» hu
or else, what I believe is more simple, a function 9„U, which is a linear function of
the lastmentioned two functions.
The relations between SU, S,U, S^U are
S,U+4&U=T. V^2iS.U.HU,
e„U+2SU=T.U*108.U.H(T.
I have not worked out the developed expressions.
The cubicontravariant is PU=
No. 65.
/ 1
bcl 1
ad — 1
ahl^l
aek + 1
o6i + 1
ftc; +1
oiy + 1
bch+l
a^ + 1
abcl
6^+1
a^ + 1
a/k+l
afg2
«/"!
V1
a/if +1
bgl +1
ot* 1
<j/» + 1
c/k + l
chj + 1
hhj + 1
ail + 1
6j^A2
c/A2
atA  2
6V 2
eU. +1
*»■ +1
Pg\
g'h^X
/A>1
ch* 1
6;7 +1
ckl +1
y« 1
cA» 1
<2;* 2
cM + l
•
fil +1
gjl +1
cM +1
/jf +2
/hi + 3
/i^i +3
yvi
/V + 2
/a> 1
/gh+3
I'A: 1
ip 1
j^ 1
//*/ + 3
j!;*i
yv 1
ghkl
/?*!
g'k+2
fjl 4
ajk 1
SA* +2
gkil
hH +2
/A»l
ghi\
y«4
hij  1
AH1
hi? +2
hP 2
/P 2
gP 2
Atf 4
jP 2
itP 2
t? 2
i« +3
Ud +3
ijl +3
V* +3
V.
P +4
±3
±3
:l=8
±7
±7
±7
±7
:fel8
<[^, 17, 0'.
144]
A THIRD MEMOIR UPON QUANTIC8.
327
No, 66.
The quintic contraTariant is QfT^
+ 1 a«6<r» + 1
a«6V + 1
a'^ftci  3
a6 V  3
o^c»A: 3
a^bcf 3
o^'c^ 3
aftc'A 3
cibcfj + 6
 6
a^cfi 3
a»6/»  3
aV + 6
aby + 6
ahc/g+ 3
a'^ftt' + 6
abcfl+ 6
rt^c^^ + 6
oAc^A + 6
+ 4
aH** + 2
ay + 2
ayi« 3
abc/h+ 3
a6cfi + 6
a^fH 3
o^ctA; + 3
abcij + 3
ahchi + 6
+ 4
ohcg^ — 6
ai^si 3
a6(^A+ 3
a6cA;^ + 6
abgi^ — 6
a6cA7 + 6
ahfgi + 9
abfi 6
ah<^ 30
 3
ab^ + 4
a6cM— 6
a6c;7 + 6
ab/ij+ 9
ac/«Z  12
ahcjk + 3
aAt«/ 12
ac«Ar» + 6
oW  12
 3
at?hk 3
a6/<7^+ 3
ahgii + 9
ab/gl24:
acfki+ 9
afc/k; + 9
acpk 6
acpj 6
ahgU + 36
+ 2
acfgh+ 3
ab/jl+ 6
ab^l 12
abgki— 12
a/V+ 3
aft^A;  6
a/V  6
ac/^A; 12
o^i^J  12
 3
acfjl + 6
abgkl+ 6
ac/A^24
o^/a^ 6
a/iH + 6
ahqhi— 12
a/*t/ + 6
oc/'Ai + 9
acfVi\2
+ 3
acgkl + 6
a6Ai^+ 6
ac^A;12
a6iZ« +18
ai'A;  6
06^/^ + 18
a/*i^A; + 3
ac/P + 18
acfkl + 36
+ 6
achU + 6
o^vA; + 3
ctcgl^— 6
ac/A:» 6
bc'h^ + 6
aMil  24
6 V + 6
ociAr^  24
acih?  12
+ 6
acijk + 3
ahP 10
acAA:t+ 9
a/^j  6
bcff^  6
acfhk+ 9
bYJ  3
a/V + 12
a/V/ + 12
+ 6
acJ^ 10
acil» + 4
oc*;* + 18
a/VA;+ 24
ici^A^  24
acA:V  12
be/ hj 12
afgil  30
aAy+ 6
+ 3
«/&V + 9
a/«A/12
af^gjl^
a/^hi+ 3
bcgjk + 9
a/VA+ 3
bcghk+ 9
a/i«J + 3
afgki  30
10
affl  12
a/y* 6
afg'k + 24
o/«P + 6
6c/«>*12
«/!;7+ 6
&cA«t  6
agki^ + 24
a/At«+ 6
+ 9
a^ki — 6
afyk^ 6
afghi + 6
qfkil  30
bcjP +18
o/^A;;  30
6cAP + 18
aAt'  6
afiP  18
12
(ighi^ — 6
a/hki+ 9
o/i^P + 12
oAr^i^ + 12
*/i^!; + 3
a/hil+ 12
bcjkl  24
a?P + 6
ai'A/ + 12
 6
agi^ +18
o^^P + 18
afijl + 12
l^gP  3
bg'k  6
fl/t; A; + 6
ft/^A+ 3
bcghj + 9
bcgh^  12
 6
ai?jl 12
o*A:»;  12
flw/^iY  30
bchH  12
6/Ai + 24
a/P  6
Vk;'^ + 12
6c// 12
bchjl + 36
+ 18
6cf + 4
^/ + 2
ahn + 6
fccA/A: + 9
V^ + 6
a^A:^ +24
b/if  18
bg%  6
6c/A  12
12
V/  3
6cA» + 4
ai^'A; + 3
b/g/ij+ 6
6^^;  30
aAi2Ar18
^^iT'A:/ + 6
Vi^ + 6
bfgp + 6
+ 2
c»A» + 2
b/hj' 6
aiP  6
ftj5^^ + 6
biY +12
aikP + 12
bghil  30
bgij' + 3
6i^A/ + 12
12
cfhp 6
6^A»^12
bchj'  6
bg^hk IS
c«AAr»  3
6cAV  6
6^/;A: + 6
(T^A^A  3
bg^jk + 6
 6
c^A«/  12
bghjk + 9
6^Ai+ 3
bghH + 24
cj^hj + 24
b/f  6
bgP  6
c/gh^+ 3
bghij  30
 6
cghjkk 9
&A«t;  6
b^l + 6
bghP + 12
c/ghk+ 6
6(/W + 12
bhi^j +24
r/^i^  30
6^'/« 18
+ 9
c/Ay — 6
bhjP + 18
6v»  6
6fiyAr; + 12
c/A^  18
6^A;7  30
6i;P +12
c/j'k + 24
bifl +12
+ 18
chjl^ +18
6/)W  12
cm + 24
bhijl  30
c/hP + 12
bgfk + 3
c/Vi'^ + 12
c^AA;+ 12
c/A«/ + 12
12
c/A:Z 12
c/i^k^  3
c^A«A;+ 3
6i/«>fc + 3
ciO'A;^  30
6Aty« + 24
c/hkl  30
cgjk^  18
c/A;A  30
 6
//Ai+ 3
/»/i!/ + 12
chH  6
bjP  6
cg^l + 6
b/P + 6
c/j/(^ + 24
chHl + 6
cgrAA* + 6
+ 12
A;"'^ + 6
/^/*»A:+ 3
cA«Z» + 6
c/h^k+ 3
cAiAr^ + 12
r/A»  6
c^A:*  6
cAyA + 6
chHk + 6
+ 3
fif  6
/hH  6
chjkl  30
cAA:*/ + 6
cij^ + 3
cA«A:; + 6
cAiA:« + 3
cAZ»  6
chkP  18
+ 6
^iW:  6
/A«^ + 6
cf^ +12
cjk"  6
ckP  6
chjk" + 3
ck'P + 6
9 A/2 +12
cjm +12
+ 6
g'hH +12
fhjklZO
/y +12
/V*'  27
/yA  27
/V 6
/T +12
/ V  6
/V^i + 18
30
i^AZ« + 6
ffl^ +12
//A«  27
/!;"'^  6
Pgil + 18
/^A«/ + 18
PiUk 12
/^A/ + 18
Pjn 24
+ 6
g'jkl + 6
i^AJfe«Z+ 6
/«;•«* 12
/%7 + 18
Z*^/*  6
/^A;* 3
Pghl + 18
fg'jk 12
/</»M + 18
+ 3
^Av7  30
9i1^  6
/^/i;7 + 36
/ghkl + 36
/v'*^ + 18
/hHj 12
/Va>  12
/^Ai; 3
fghH + 18
 6
gipk + 3
A«tA:Z + 6
/hip  12
/^•^12
fqhil + 36
/Ai^  6
PP 24
/^•^  6
fghP  54
+ 12
<Z;7»  6
Ai/jfe« + 3
;!rP 24
/A«i/ + 18
/W^ 3
ffkl + 18
//A?  6
/(;•'/ + 18
/7iA/  12
 6
h%y +12
MP  6
^hkl + 18
/Ai/A 3
fgl? 18
i^AA:>  6
fghikZ
g'k' +12
/Ai;7 12
+ 6
i7/» + 6
iA«P + 6
!/!;^  6
/A/» 18
yAt^i 12
gh^ki 12
fgkP  6
g%ki\2
/i/*A+18
ghHl + 18
j5^p  6
/i/P  6
ghkP  6
fhS"  6
fkP 24
/;/» +48
ghijk  3
^F +12
fiei  6
gjm + 18
/AiP  6
^A«i»  6
fkH 24
^/i^» 18
(^/lA^^i  12
ghk^  12
A\^ + 12
fijkl + 36
ghiP  6
ghkil 12
i^'/t/^  6
gk^P 24
i^>ti7»  6
hHP 24
/^ + 12
^(/A/ + 36
gij^ + 18
AVy  6
/i«A:i«  6
AV + 12
A/^ + 12
giMl + 18
gP + 12
^A/* +48
hip  6
fikiP  6
Ai«P 24
hijkl + 36
Ai^A;; + 18
At^/ +18
AW 24
V*A;Z +18
t/A:*/ +18
i^jkl +18
ifJ^ 27
t^A* 27
i^j'k 27
hiP +48
p + 12
kl* + 12
t7* + 12
JAjP 18
ikP 18
iA/»  18
Ai!/A + 18
ijkP 54
Z» 24
'\
^{t V (r
dbl45
14o
i:282
=b282
:i^282
:1=282
:t282
282
db486
328
A THIRD MEMOIR UPON QUANTICS,
[144
No. 67.
The reciprocant is FU={*^^, 17, f)' =
^
W +1
bc/i  6
W» +4
cP +4
aV +1
acgj— 6
o^ +4
cf +4
pV3
f^.
a»6» +1
abhk — 6
oifc* +4
6A» +4
h^Jc" 3
^r.
r»^
9 *
 6
aegh + 6
ag'l
chf
if
+ 12
+ 18
24
12
+ 6
+ 12
12
abfh
ahkl
afie
hhH
bhjk
fh^k
hiei
 6
+ 6
+ 12
12
24
+ 18
+ 6
+ 12
12
iS
hi?k
hcfg
bcil
qfH
 6
+ 6
+ 12
12
24
c/H + 18
ft'l +12
«  12
iji*.
a V  6
abhl + 12
abjk + 6
a/hk + 18
ak'l
bhy
24
12
12
+ 12
+ 6
if.
l^cg
btifl
bcik
hfgi
bH'l
cPk
P9
A^k
 6
+ 12
+ 6
+ 18
24
12
12
+ 12
+ 6
fif".
ac»A  6
ciegl + 12
acij + 6
agH 12
cghj + 18
cpi 24
^A 12
9^1 +12
^i" + 6
±9
9
9
54
54
54
54
db54
54
n*i*.
W
W.
i?»f*.
i*e*.
ev^>
T'f.
f*?.
?»i».
a V + 6
o^V + 6
6c«A + 6
a»6» + 6
6»c/ + 6
ac'k
+ 6
a'be  2
a6«c  2
oW  2
aV + 9
ab/l  12
6c^/ 12
ay + 9
by + 9
acfg
 6
aVi  18
a6/t + 6
ac/» + 6
achl  12
abik — 6
bcij  6
a5^/* — 6
fcc/A  6
acil
12
abgj + 6
a/»  4
at'  4
oc; A;  6
a/»A; + 12
6^ +12
abjl 12
6dW 12
agp
+ 12
achk + 6
6«a; 18
ftca; + 6
a/gj 18
^i" + 9
c»A:» + 9
a/A/  36
b/gl 36
c»A«
+ 9
afgh + 18
6cA)k+ 6
V  4
o^A; +12
6/A;  18
c/y +12
a/jk  18
b/ij 18
cJP
+ 12
afjl +36
bfgh + 18
c'Ait 18
aghi  18
bghk  18
(2/5^A;  18
agk" +12
6^ 18
cghl
36
agkl  48
^^gf/ +36
c/flrA + 18
a^^ +48
bhH +12
c/hi  18
atAifc  18
6Ai» +12
cgjk
18
ahU +36
bgld + 36
cfjl 48
aijl 36
6AZ« +48
c//» +48
oA^ +48
biP +iS
chij
18
aijk +18
6AiZ  48
<^A^ +36
cAV +12
6;A^ 36
o^Z 36
bkp +12
c/*» +12
cjP
+ 48
a/* 32
bijk +18
cijk +18
or +12
/W  3
/y  3
^A'A:  6
/!/ +12
m
 6
6;'  4
6/" 32
cAt/ +36
i^A»  3
/AA;Z  24
/gil 24
A'j +36
/^gk + 36
fk
+ 12
cA»  4
cP  4
c/» 32
5rA;7 24
^k" +36
/»!;  6
A«t +12
/•Af  6
fhi
+ 36
/y 36
/»AZ + 12
/^Z +12
Sfk  6
^/fc* +12
gi'k +36
h^P 12
PP 12
g'P
12
ghH +12
Pjk  36
^V +12
Atf +36
hik"  6
Ai» + 12
A;;M 24
/a/  24
gijl
24
(^Ajife + 12
/i7ife» 36
g'ki 36
fP 12
*»Z» 12
*•/« 12
/*»  3
Pk"  3
»y
 3
AV 36
hjP +24
y»« +12
fhki +12
y^ +24
tib"Z +12
^At« 36
^ +24
Pjl +12
135
135
:i^l35
135
dbl85
185
180
11x180
rfelSO
U]
A THIRD MEMOIR UPON QUANTIC8.
329
12
30
12
24
18
12
24
24
12
18
60
12
66
12
48
60
V*^'
ac/j
acgk
achi
agtl
ai?j
egh^
chjl
cpk
fhl
fjk
ghij
jHl
12
12
+ 30
24
+ 24
+ 12
18
18
+ 12
+ 24
12
+ 60
12
66
48
+ 60
i^fi?.
abgk
abhi
abij
abP
aph
afkl
aiJ^
hgh^
hhjl
bj^k
fhH
fhjk
gW
h^ki
hkP
12
12
+ 30
24
18
+ 12
+ 24
+ 24
+ 12
18
+ 60
66
12
12
48
+ 60
wr.
cibci
aep
a/t^
hcgh
hcjl
hfl
igij
cfhl
cfjk
cg^
chik
ckP
fgh%
gikl
hn
i^jk
+ 6
12
+ 6
24
+ 36
+ 12
30
60
+ 12
36
+ 54
+ 24
24
+ 60
+ 78
96
+ 72
12
48
66
+ 48
7»r»^
abcQ
abg^
acfh
ackl
afgl
a/ij
ahi^
agik
ail^
chH
chjk
fghj
m
g'hk
ghH
ghJI?
gjkl
hijl
ij^k
+ 6
12
24
+ 36
60
+ 54
36
+ 12
+ 24
+ 6
+ 12
30
+ 78
48
24
+ 60
96
+ 72
12
66
+ 48
f*ft7».
abck
ah/g
Ml
api
a/ik
hch^
hghl
bgjk
bhij
bp
cm
fghJc
fhH
fhJ^
fi^i
gm
hikl
+ 6
24
+ 36
+ 12
30
12
36
60
+ 54
+ 12
+ 24
+ 6
+ 60
+ 78
24
96
12
48
+ 72
66
+ 48
^17^'.
abcf
abi^
afH
bchl
bcjk
bfk
bghi
bffP
bijl
e/hk
cm
Ml
fhil
fijk
fP
giJ^
hi^k
ikP
+ 6
12
+ 6
+ 36
24
+ 54
36
+ 12
+ 24
60
30
+ 12
66
48
12
+ 72
+ 78
+ 48
+ 60
24
96
iy»rt«.
ahcg
acfl
acik
ofgi
aiH
bcp
Vi
cghJc
chH
chP
cjkl
fail
g'kl
ghil
gijk
hiy
+ 6
+ 36
24
30
+ 12
12
+ 6
+ 12
+ 54
36
+ 24
60
66
+ 72
24
48
12
+ 78
+ 48
+ 60
96
f'^^.
ahch
abgl
abij
a/gk
afhi
afP
aikl
hghj
bfl
ch^k
fhjl
m
ghkl
h^il
hijk
hP
jkP
+ 6
+ 36
24
12
36
+ 12
+ 54
+ 24
60
30
+ 12
+ 6
66
12
+ 60
+ 72
24
48
+ 78
+ 48
96
wr».
abcl
aJbgi
acfk
oPg
afxl
w?k
bchj
bgVi
bgjl
bip
cfli^
chkl
\fgh.l
/jp
g'i^
ghkl
gkP
h^?
hiP
ijkl
I*.
}22
db222
=1=222
:i=408
:Jb408
±498
±408
408
408
24
6
6
30
48
30
6
30
48
30
30
48
30
24
+ 108
114
 114
+ 24
+ 24
114
+ 24
+ 24
+ 24
+ 108
 48
±558
+
+
+
+
+
+
+
+
+
+
The preceding Tables contain the complete system [not so] of the covariants and
ntmvariants of the ternary cubic, i.e. the covariants are the cubic itself U, the
lartinvariant 8, the sextinvariant T, the Hessian HU, and an octicovariant, say SU;
e contravariants are the cubicontravariant PU, the quinticontravariant QUy and
e reciprocant FU.
The contravariants are all of them evectants, viz. PU is the evectant o{ 8, QU
the evectant of T, and the reciprocant FU is the evectant of QU, or what is the
me thing, the second evectant of T.
The discriminant is a rational and integral function of the two invariants; repre
Qting it by E, we have i2 = 64 8^1^.
If we combine U and HU by arbitrary multipliers, say a and 6)8, so as to form
e sum aU +6/3HU, this is a cubic, and the question arises, to find the covariants
d contravariants of this cubic : the results are given in the following Table :
7^6^HU
{aU^6/3HU)
aU^e^HU.
(0, 2S, T,
+ (1, 0,125,
No. 68.
8S«3[a, /3yu
C. II.
42
aso
A THIRD MEMOIR UPON QUANTIOS.
144
P(aU+6^HU)= (1. 0, 125, iT^a, ^yPU
4(0,1. 0,  'iSlia. ^y QU.
(0, 60.Sf, SOT, 0,1 20T8.  242* + 5765* \a, fiy P U
+ (1, 0, 0, lor, 2405*, 2^T8\a, ^yqu.
(5, T, 245', 4^5, 2*485»3^a, ^y.
{T, 965", 6075, 202*. 2402*5*.  482*5 + 46085*.  82* + 5762'S»'J[a, jS/.
[(1,0, 245,82', 4&S'\t, ^y\*R.
(1,0. 245, 87,  4»S''$_a, 0y FU
+ (0. 24, 0, 0, isT^cL, fiy(PUy
+ (0. 0. 24, 0, 9QS\a, fiyPU.QU
+(0, 0, 0. 8. 03[a, fiy.{QUy.
We have, in like manner, for the covariants and contravariants of the cubic
GttPU+l3QU, the following Table:
5 (aU + 6fiHU) =
T(aU+6fiHU) =
R(aU+Q^HU) =
F(aU+6fiHU) =
No. 69.
6(xPU+fiQU =fi<iPU + PQU.
H(6aPU + fiQU) = (2T. 485*, 182'5, 2* + 165»'5a, ^yPU
+ (85, T,  85«,  T8 \<i, ^y Q U.
P (6aPU+ ffQU) = (325*, 122'5, 2* + 325', 42'5'3^a, /9)» U
+ (47, 965'. 122'5, 2*  325* 3[«, ^S)* HU.
Q(QaPU + 0QU)= r + 3842'5»,
+ 1202*5 + 7680 5*,
+ 102* +320075*,
+ 4802*5»,
+ 302*5,
1^+ 12^  242*5* + 6125"
+ ( 242* +4608 5>, "^
+ 19202* 5*.
+ 4802*5,
+ 302'* + 19202'5»,
+ 1202"5» + 7680 5*,
62"5 + 76875*
6 a, ^yU
)
a, fiy HU.
44]
A THIKD MEMOIR UPON QUANTIC8.
331
8(6aPU + ^QU) =
T(6aPU+l3QU) =
(+ ir» +192 S", '
+ 128TS\
+ 18T*S + 384 S\ k a, fi)\
+ IT* + 64^/8",
t+ 5r»S« 64 Sf" J
' 8r» + 4608rS»,
+ 1920r»iS» + 73728 /S*.
"i
+ 360r»<Sf + 384002'/S',
+ 2or* + sgeor'/S",
+ 84or'/S»+ lesoTS',
+ S6T*S + SSiPS* + 24576 S',
{+ 12"  40r'/S»+ 2560rS* ;
+ /3QL0 = [(48&', Sr, 965*,  247/8,  2"  16/Sf "go, /3)']»i?.
f/8Qfr) = ( 192S, 322* ,384 S' , 96T8
+ ( , 512 iS», 1922* /Sf , 24r'/Sr
+ (1344S', 3522'«S, 24r'1152S», 2882'S»
+ (48 2'. , 2882' S , 247' + 1536/8*,
6 a, /3)'.
47' 64fi^3[a, /3)*eir
2'' 5a, /9y. U*
20T'8 + 64/S*3ia, /9)« U. HU
The tables for the ternary cubic become much more simple if we suppose that
16 cubic is expressed in Hesse's canonical form; we have then the following
ible:
U
S
T
R
HU
SU
e.u =
No. 70.
^ + y* + •** + Qlxyt.
l+l*.
1 _ 20?  81*.
 (1 + 8P)».
I* (x* + y* + z*)  (1 + 21') xyz.
a + 8J*y {fi> + z'a^+ ai'y*)
+ (9l*)(x' + y* + ^y
+ {2l ol*  20r) (a;» + y» + ^») xyz
+ ( 151*  781' + 121*) ai'y'z*.
4 (1 + 8l*y (y*!* + 1^*0^ + a?y*)
+ (lW4>l*)(ai> + y* + z»y
+ (4/ + lOOi* + 112P) (a!» + y» + «») xyz
+ (48f + 552P + 48f) !d*y*z*.
42—2
332
A THIRD MEMOm UPON QUANTICS.
[144
^,,U=  2 (1 + Sl'fifz' + 2^0^^ a^f)
+ (iioz»)(^ + y' + 8*)»
+ (6i  180Z* + 96^) (a^ + y» + ^) ayyz
+ (6f»  624P  192i«) a^fz\
PU =Z(p + i?* + (r) + (l + 4?)^7C.
QU =(l10P)(f* + if + (r)6?(5 + 4i»)fi;?.
FU =  4 (1 + 8P)(i;»?» + ?»p + fi7»)
 24Z (1 f 2f») fi7»^,
to which it is proper to join the following transformed expressions for ©CT, ©,i/, B^^IT,
viz. 8J7 = (1 + 8Z»y (y»^ + ^a^ + a;»y»)
+ (2Z  5Z*) 17 . iTIT
+ (3i» )(irco.
e,J7 = 4 (1 + 8Z»)» (y»^ + ?»a^ + «»y»)
+ (16i + 4/* )U.HU
+ (12i« )(^^'.
e,,ir= 2(1 + 8P)«(y»^ + ?»a:» + a^y»)
+ (6Z )i;'.irfr
+ (6Z» )(HU)\
The last preceding table affords a complete solution of the problem, to reduce a
ternary cubic to its canonical form.
[I add to the present Memoir, in the notation hereof (a, 6, c, /, g^ A, i, j, it, V^x, y, zf
for the ternary cubic, some formulae originally contained in the paper "On Homo
geneous Functions of the third order with three variables," (1846), but which on account
of the difference of notation were omitted from the reprint, 35, of that paper.
Representing the determinant
cuc + hy + jz,
hx + ki/ + bz,
jx + iy + gz,
hx + ky + bz, jx +li/ + gz, f
kxfby+fz, Ix +Jy + tz, tf
Ix ^rfy + iz, gx + %y + cz, f
A THIRD MEMOIR UPON QUANTICS.
333
(A, B, C, F, 0, E^x, y, zf
alues of 4, B, (7, F, 0, H (equations (10) of 35) are
jover writing
lat
B
G
H
^
Cf
^
 p
• bi
 r
 i*
be
fi
fg
+ ek
2U
ki
+ bg
2fl
ag
 J'
hi
 p
 f
• ■
V
+ eh
2gl
ea
 SO
gh
+ M
2jl
ok
 A»
bh
 P
¥
+ bj
2kl
+ a/
2hl
ah
 hk
2M
2al
2kl
2V"
2gl
2t>
P
+ gk
 hi
gh
— ai
 «/
2hl
2jk
2/k
2U
2*;
2/&
ki
 hg
P
+ hi
Jj
 gi^
¥
 y
2jl
2gh
2/7
2ki
2^
2el
f9
 ek
• •
 cA
p
 gk
 hi
FU^'
a, k, g, I, j, h,
h, h, i, f, I, k,
j, f, e, i, g, I,
2f . . . K V
A, B, C, F, G, H
FU = A& + Eo + Cc + 2Fi+ 2(?g + afHi,
334
A THIRD MEMOIR UPON QUAKTICS.
[14
then the values of a, b, c, f, g, h (equations (13) of 35) are
(?e
(W
eni
w
^c
2c/
2t«
2bi
2/»
 6c
2q;
2g>
2a5r
2/
si
— ca
2hh
2P
2aA;
2A»
Aii;
 ab
6y
2cA
4A;
4a/
2/
2a^
3at
^2jl
ca
 93
Afk
46/
Shi
6; A;
2a/
a6
2A:«
26A
2kl
 ¥
4^i
4c/
8//
26^
3cA;
2i/
6c
2»'
2c/
Ski
26;
Ahj
4a/
2A»
2a^
ab
 hk
3a/
2«
8i/
2cA;
4/A;
46/
36<7
2/1
 ki
ca
2/«
26*
6e
 f*
Agi
4c/
Sjl
Qgh
2ai
3cA
2gl
• m
 «
2p»
2c;
6Ai
+ 6/i
4(7A;
8/»
2asr
2/
2aA;
2A*
Aal
Ahj
+ 2AZ
3a/
+ 2j/
3a»
26t
2/»
6^'
4A»
26A
¥
+ 2kl
3bj
46/
4/*
Id
+ 2/1
Sbg
2c/
2i^
2c;
2<^
6gk
SP
m m
V
+ 2gl
3ch
+ 2t/
3ck
id
igi
2/i
26c
4cA
+ 4(7/
8t;
46;
+ 4A:/
8/t/
AP
+ 2hi
Sgk
6/1
 hg
ifg
 dc
4db
44t/
2ca
4a/
+ 4A/
8iA;
7gh
 6jl
— ai
4P
+ 2y*
8A»
7v
6gl
— ch
•
46(7
+ 4//
8A:t
Aai
+ Ajl
+ Sgh
2M
2a6
7jk
ehi
 ¥
7A/
4P
+ 2y*
+ 2Ai
8;5f
144]
A THIRD MEMOIR UPON QUANTIC8.
335
Also if the discriminsat be written
K(U) =
a
k
9
I
•
3
h
h
b
■
I
f
I
k
•
•
c
•
X
9
I
^
m
05
%
%
n
n
iS
I
s
%
m
aj
I
OD
{
as
%
then the values of a, 33, ®, Jp, ffi, ^, I, 2I» ^> 'I (equations (20) of 35) are
a = a% + 2hjl " aP  gh^  j^k,
33 = 6iA + 2/ikf  6P  A/«  Jfc^i,
(!C=(2/' +25n7  cfi  /^r^  i^;,
3JF = 6cA + K;  cA:» +2gfk2bgl+fP ^pj  /lA,
3ffi = ca/ + cjk  ai« + 2A5ri  2cAi + 5rP ^r'A?  53/;
3^ = a65r+ aJfci  6f +2fhj ''2afl + hl^ hH  AJfc^r,
3 I = 6c; + c/t  65r» + 2%  2ckl hjl^  i»A  fij,
33J = CO* + oflr/ cA> + 2ijh  2aa + kl^ ff  a/Jfc,
3!a = aW + bhg" ap + 2j*/  2hjl^il^ k^g  hki,
61 = ofcc + 3^A + Sijk +2/*  a/i bgjchk  2Z5fA;  2iAi  2ljj,
The equation JST ( IT') = i2 = 64iS* — T* would however afford a perhaps easier formula for
the calculation of the discriminant.]
336
[145
145.
A MEMOIR UPON CAUSTICS.
[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the
year 1857, pp. 273—312. Received May 1,— Read May 8, 1856.]
The following memoir contains little or nothing that can be considered new in
principle; the object of it is to collect together the principal results relating to caustics
in piano, the reflecting or refi::acting curve being a right line or a circle, and to
discuss, with more care than appears to have been hitherto bestowed upon the subject,
some of the more remarkable cases. The memoir contains in particular researches
relating to the caustic by refraction of a circle for parallel rays, the caustic by
reflexion of a circle for rays proceeding from a point, and the caustic by refiuction
of a circle for rays proceeding from a point; the result in the last case is not
worked out, but it is shown how the equation in rectangular coordinates is to be
obtained by equating to zero the discriminant of a rational and integral function of
the sixth degree. The memoir treats also of the secondary caustic, or orthogonal
trajectory of the reflected or refracted rays, in the general case of a reflecting or
refracting circle and rays proceeding from a point; the curve in question, or rather
a secondary caustic, is, as is well known, the Oval of Descartes or 'Cartesian': the
equation is discussed by a method which gives rise to some forms of the curve which
appear to have escaped the notice of geometers. By considering the caustic as the
evolute of the secondary caustic, it is shown that the caustic, in the general case of
a reflecting or refracting circle and rays proceeding from a point, is a curve of the
sixth class only. The concluding part of the memoir treats of the curve which, when
the incident rays are parallel, must be taken for the secondary caustic in the place
of the Cartesian, which, for the particular case in question, passes off to infinity. In
the course of the memoir, I reproduce a theorem first given, I believe, by me in the
Philosophical Magazine, viz. that there are six different systems of a radiant point
145] A MEMOIR UPON CAUSTICS. 337
and refracting circle which give rise to identically the same caustic, [see post, xxviii].
The memoir is divided into sections, each of which is to a considerable extent in
telligible by itself, and the subject of each section is for the most part explained
by the introductory paragraph or paragraphs.
I.
Consider a ray of light reflected or refracted at a curve, and suppose that ^, 17
are the coordinates of a point Q on the incident ray, a, fi the coordinates of the
point Q of incidence upon the reflecting or refracting curve, a, b the coordinates of
a point N upon the normal at the point of incidence, x, y the coordinates of a
point q on the reflected or refracted ray.
Write for shortness,
(6y3)(fa)(aa)(,,^)= VQQN,
(a  a) (f  a) + (6  /S) (i,  /S) = U^QN,
then ^QGN is equal to twice the area of the triangle QON, and if f, 17 instead of
being the coordinates of a point Q on the incident ray were current coordinates, the
equation VQGN=0 would be the equation of the line through the points G and N,
Le. of the normal at the point of incidence ;* and in like manner the equation
□QG^i\r = would be the equation of the line through G perpendicular to the line
through the points G and AT, ie. of the tangent at the point of incidence.
We have
and therefore identically.
NG =(aa)» + (6)8)«,
W 'QG'='^QG^+ dqgn\
Suppose for a moment that <f> is the angle of incidence and <f>' the angle of reflexion
or refraction; and let fi be the index of refraction (in the case of reflexion /Lt = — 1)^
then writing
(6y8)(a;a)(aa)(y;8)=V5rGi\r,
and
we have
56« = (a:a)« + (yi8)^
. ^ VQGN . ^, VqGN
''''''' = NGTGQ' ^^^*=^GTG^'
and substituting these values in the equation
sin* (fyfi^ sin* 0' = 0,
C. II.
43
338 A MEMOIR UPON CAUSTICS. [l45
we obtain
^* V^GN'  fi' Q^VqGN^ = 0,
an equation which is rational of the second order in x, y, the coordinates of a point
q on the refracted ray; this equation must therefore contain, as a factor, the equation
of the refracted ray; the other factor gives the equation of a line equally inclined
to, but on the opposite side of the normal; this line (which of course has no physical
existence) may be termed the false refracted ray. The caustic is geometricallt/ the
envelope of the pair of rays, and for finding the equation of the caustic it is
obviously convenient to take the equation of the two rays conjointly in the form
under which such equation has just been found, without attempting to break the
equation up into its linear factors.
It is however interesting to see how the resolution of the equation may be
effected; for this purpose multiply the equation by Nffi, then reducing by means of
a previous formula, the equation becomes
C^^GN' + a^Qlt)VQGN^  A'^W^^ + aQGN')'7^N' = 0,
which is equivalent to
and the factors are
Vj(?iV\//A«DQGy + (M— l)VQGi7" TDqGN.VQGN^O;
it is in fact easy to see that these equations represent lines passing through the
point G and inclined to GN at angles ± <t>\ where 0' is given by the equations
sm<f> = fi sin <f>\
and there is no difficulty in distinguishing in any particular case between the refracted
ray and the false refracted ray.
In the case of reflexion /4* = — I, and the equations become
^qGN. DQGN+ UqGN . VQGN^O;
the equation
VqGN, DQGNOqGN. VQGN^O
is obviously that of the incident ray, which is what the false refi*acted ray becomes
in the case of reflexion ; and the equation
VqGN . DQGNhDqGN. S/QGN =
is that of the reflected ray.
145] A MEMOIR UPON CAUSTICS. 339
11.
But instead of investigating the nature of the caustic itself, we may begin by
finding the secondary caustic or orthogonal trajectory of the refracted rays, i.e. a curve
having the caustic for its evolute; suppose that the incident rays are all of them
normal to a certain curve, and let Q be a point upon this curve, and considering
the ray through the point Q, let G be the point of incidence upon the refracting
curve ; then if the point G be made the centre of a circle the radius of which is
fjT^ . GQ, the envelope of the circles will be the secondary caustic. It should be
noticed that, if the incident rays proceed from a point, the most simple course is to
take such point for the point Q. The remark, however, does not apply to the case
where the incident rays are parallel ; the point Q must here be considered as the
point in which the incident ray is intersected by some line at right angles to the
rays, and there is not in general any one line which can be selected in preference
to another. But if the refracting curve be a circle, then the line perpendicular to
the incident rays may be taken to be a diameter of the circle. To translate the
construction into analysis, let f, rj be the coordinates of the point Q, and a, 13 the
coordinates of the point G, then f, 17, a, ^8 are in efifect functions of a single
arbitrary parameter ; and if we write
then the equation
where x, y are to be considered as current coordinates, and which involves of course
the arbitrary parameter, is the equation of the circle, and the envelope is obtained
m the usual manner. This is the wellknown theory of Gergonne and Quetelet.
III.
There is however a simpler construction of the secondary caustic in the case of
the reflexion of rays proceeding from a point. Suppose, as before, that Q is the
radiant point, and let G be the point of incidence. On the tangent at G to the
reflecting curve, let fall a perpendicular from Q, and produce it to an equal distance
on the other side of the tangent; then if q be the extremity of the line so produced,
it is clear that g is a point on the reflected ray Gq, and it is easy to see that
the locus of J is the secondary caustic. Produce now QG to a point Q' such that
GQ' = QGy it is clear that the locus of Q' will be a curve similar to and similarly
situated with and twice the magnitude of the reflecting curve, and that the two
curves have the point Q for a centre of similitude. And the tangent at Q' passes
through the point 5, Le. q is the foot of the perpendicular let fall from Q upon
the tangent at Q\ we have therefore the theorem due to Dandelin, viz.
43—2
340 A MEMOIR UPON CAUSTICS. [l45
If rays proceeding from a point Q are reflected at a curve, then the secondary
caustic is the locus of the feet of the perpendiculars let fall from the point Q upon
the tangents of a curve similar to and similarly situated with and twice the magni
tude of the reflecting curve, and such that the two curves have the point Q for a
centre of similitude.
IV.
If rays proceeding from a point Q are reflected at a line, the reflected rays will
proceed from a point q situate on the perpendicular let fall from Q, and at an equal
distance on the other side of the reflecting line. The point q may be spoken of as
the image of Q; it is clear that if Q be considered as a variable point, then the
locus of the image q will be a curve equal and similar but oppositely situated to
the curve, the locus of Q, and which may be spoken of as the image of such curve.
Hence it at once follows, that if the incidental rays are tangent, or normal, or indeed
in any other manner related to a curve, then the reflected rays will be tangent, or
normal, or related in a corresponding manner to a curve the image of the first
mentioned curve. The theory of the combined reflexions and refractions of a pencil
of rays transmitted through a plate or prism, is, by the property in question, rendered
very simple. Suppose, for instance, that a pencil of rays is refracted at the first
surface of a plate or prism, and after undergoing any number of internal reflexions,
finally emerges after a second refraction at the first or second surface; in order to
find the caustic enveloped by the rays after the second refraction, it is only necessary
to form the successive images of the first caustic corresponding to the different reflexions,
and finally to determine the caustic for refraction in the case where the incident
rays are the tangents of the caustic which is the last of the series of images; the
problem is not in effect different from that of finding the caustic for refraction in
the case where the incident rays are the tangents to the caustic after the first re
fraction, but the line at which the second refraction takes place is arbitrarily situate
with respect to the caustic. Thus e.g. suppose the incident rays proceed from a
point, the caustic after the first refraction is, it will be shown in the sequel, the
evolute of a conic; for the complete theory of the combined reflexions and refractions
of the pencil by a plate or prism, it is only necessary to find the caustic by refraction,
where the incident rays are the normals of a conic, and the refracting line is arbitrarily
situate with respect to the conic.
V.
Suppose that rays proceeding from a point Q are refracted at a line; and take
the refracting line for the axis of y, the axis of x passing through the radiant point
Q, and take the distance QA for unity. Suppose that the index of refraction /a is
put equal to r. Then if ^ be the angle of incidence and <(/ the angle of refraction.
145]
A MEMOIR UPON CAUSTICS.
341
we have sin <f>' = k sin <t>, and the equation y — x tan <f/ = tan <t> of the refiracted ray
becomes, putting for ^' its value,
y — 7==== X — tan = 0.
^ VlJfc«sin«<^ ^
Differentiating with respect to the variable parameter and combining the two equations,
we obtain, after a simple reduction,
kx='
{lk' sin* <^)*
cos*^
A'* sin'
*'^""~ cos»0 '
where i' = Vl — A» ; hence eliminating,
(Aa:)*(A'y)*=l,
which is the equation of the caustic. When the refraction takes place into a denser
medium k is less than 1, and k'^ is positive, the caustic is therefore the evolute of
a hyperbola (see fig. 1); but when the refraction takes place in a rarer medium k
is greater than 1, and A'* is negative, the caustic is therefore the evolute of an
ellipse (see fig. 2). These results appear to have been first obtained by Gergonne.
The conic (hyperbola or ellipse) is the secondary caustic, and as such may be obtained
as foUowa
VI.
The equation of the variable circle is
aj" 4 (y  tan <^)»  A:" sec" = ;
or reducing, the equation is
^ ^y*  2y tan <^ + A'« tan«<^ ifc« = :
whence, considering tan as the variable parameter, the equation of the envelope is
A:'«(aj« + y«*»)y» = 0,
that is,
jf'ar'  Jfcy  A^^jfc'" = 0,
A HiaiOIR UPON CAUSTICS.
is the equation of the secondary caustic, or conic haviog the caustic for its evol
The radiant point, it is clear, is a focus of the conic
VII.
Let the equation of the refracted ray be represented by
Xa:+Yy + Z = 0,
ve have
from which we obtain
Vl  ft» sinV
3> F'~2»
for the tangential equation of the caustic ; or if we represent the equation of
refracted ray by
Zx+Yyk = 0.
then we have
X' 7' f
for the tangential equation of the caustic
Fig. 1.
Fig. 2.
m
A HBUOtR UPON CAUSTICS.
vin.
If a ray be reflected at a circle; we may take a, 6 as the coordinates of the
centre of the circle, and supposing as before that f, 17 are the coordinates of a point
Q in the incident ray, a, j8 the coordinates of the point of incidence, and x, y
the coordinates of a point q in the reflected ray, the equation of the reflected ray,
treating 3;, y as current coordinates, is
((i/3)(«.)(o.)(y»n(a.)(f.) + (i«(lS)l
+ !(««)(«.) +(i«(y(J))Ki»(t.)(a.)(,^)).0.
Write for shortness,
*,..('»(««)(»a)(y«.
T,., .(ai.)(»a) + (iS)(yffl,
and similarly for Nf^^a &c. ; the equation of the reflected ray is
Suppose that the reflected ray meets the circle agun in Q' and undergoes a
second reflexion, and let jb', y* be the coordinates of a point q' in the ray thus twice
reflected. We see first ((?' being a point in the first reflected ray) that
Again, considering G as a point in the my by the reflexion of which the second
reflected ray arises, the equation of the second reflected ray is
uid from the form of the expressions Nq,a, '^q.a ^^ ^ clear that
he equation for the second reflected ray may therefore be written under the form
r reducing by a previous equation, we obtain finally for the equation of the second
iflected ray,
td in like manner the equation for the third reflected ray is
1 n on, the equation for the last reflected ray containing, it will be observed, the
tdinates of the radiant point and of the first and last points of incidence (the
^dinates of the liist puitit of incidence can of course only be calculated &ora those
: radiant point and thu first point of incidence, through the coordinates of the
iate pqinU of incidence), but not containing explicitly the coordinates of any
intermediate points of incidence. The form ia somewhat remarkable, but the
I Is nal^ the same with that obtained by simple geometrical considerations, as
^
344
A MEMOIR UPON CAUSTICS.
[145
IX.
Consider a ray reflected any number of times at a circle; and let G^G, be the
ray incident at G, and GG' the last reflected ray, the point at which the reflexion
takes place or last point of incidence being G. Take the centre of the circle for
the origin, and any two lines Ox, Oy through the centre and at right angles to each
other for axes, and let Ox meet the circle in the point A, Write
/.AOG, =^0, Z.xG,G, = iro,
ZAOO =0, ZxGG' =i/r,
z GoG,0 = <l> ;
then the radius of the circle being taken as the centre of the circle, the equation
of the reflected ray is
y — sin ^ = tan '^ (a? — cos 0) ;
and if there have been n reflexions, then
=^o + w(7r2<^) = ^o +n7r2n<^,
•^ = '^o — 2n0,
and therefore the equation of the reflected ray is
y cos (iiro  2n0)  x sin (i/to  2n<^) + ()** sin (^o  ^o) = 0.
X.
If a pencil of parallel rays is reflected any number of times at a circle, then
taking AO for the direction of the incident rays, we may write ^o = 0, '^o = 'n", and
the equation of a reflected ray is
X sin 2n<l> + y cos 2n^ = (—.)'* sin ^ ;
145] A MEMOIR UPON CAUSTICS. 345
differentiating with respect to the variable parameter, we find
X cos 2n^ — y sin 2n^ = (— )** 5 cos ^ ;
and these equations give
X = ^^ I (2n + 1) cos (2n  1) <^  (2n  1) cos (2w + 1) ^l ,
y = ^^ I  (2n + 1) sin (2n  1) + (2n  1) sin (2w + 1 ) </>! ,
which may be taken for the equation of the caustic; the caustic is therefore an
epicycloid: this is a wellknown result.
XL
If ra3r8 proceeding from a point upon the circumference are reflected any number of
times at a circle, then taking the point A for the radiant point, we have ao = 0,
^^ = 'jr — <f>, and the equation of a reflected ray is
a;sin(2n+ l)0f y cos (2n + l)0 = (—)** sin <f>\
differentiating with respect to the variable parameter, we find
^cos(2r{ + 1)0 — y sin(2n+ 1)0 = ( — )*» sin0;
and these equations give
()» ( )
a? = o '^ < (n + I)cos2n0— wco8(2n + 2)0 ,
(Y ( )
y^Zi j(w + l)8in2ii0 + ncos(2n + 2)0
which may be taken as the equation of the caustic ; the caustic is therefore in this
case also an epicycloid: this is a wellknown result.
XII.
CJonsider a pencil of parallel rays refracted at a circle; take the radius of the
circle as unity, and let the incident rays be parallel to the axis of x, then if 0, 0'
be the angles of incidence and refraction, and /n or t be the index of refraction, so
that sin 0^ = A; sin 0, the coordinates of the point of incidence are cos 0, sin 0, • and
the equation of the refracted ray is
y — sin = tan (0 — 0') (x — cos 0),
le.
cos (0 — 0O(y — sin 0) = sin (0 — if}') (x — cos 0),
c. II. 44
346 A MEMOIR UPON CAUSTICS. [145
or
y cos {<f> — 0') — a? sin (^ — if>) = sin 0',
which may also be written
(y cos <^ — a? sin 0) cos 0' + (y sin + a? cos — 1) sin ^' = 0.
Writing k sin ^, vT— A;* sin' j> instead of sin if/, cos <f>\ and putting for shortness
y cos — a? sin = F,
y sin + a? cos <^ = X,
A? sin _ .
Vl  ^> sin* <^
the equation of the refracted ray becomes
F+4>(Z1) = 0;
and differentiating with respect to the variable parameter 0, observing that
dF_ y dX _ «.
d<^ ' d(f>
d4>_ icos<<^ _ cot^ .
# " (1  A:> sin«^)* "" 1^ A:* sin* if> '
we have
V 1  if 81U' / '
4>
and the combination of the two equations gives
y.^_^(ljA^8in»^
<Pcot</>l '
^ _ 4> c ot (^ — A;* si n'
4>cot<^l " •
and we have therefore
TT ^ , v • ^ Ar'sm'^^^coti^l) , . . , .
V=xcos6 + XsinA= ^\ , — r ^ = A:^ sin* 6,
^ ^ ^ 4>cot<^ — 1 ^
* ( . — 7  A;* sin* 6 I  A;* sin* 6 cos 6
a? = Xcos6— Fsm A= ^^ g ; — ;
^ ^ 4> cot <^  1
i.e.
_ ^ (1 — A;* sin* <b) — A:* sin* <^ cos ^
~ <P cos — sin '
145] A MEMOIR UPON CAUSTICS. 347
or multiplying the numerator and denominator by (1 — A:* sin* <^) (4> cos ^ + sin <^), the
numerator becomes
(1  A:* sin* <^) {** cos <^ (1  A;* sin* <^)  A:* sin* <^ cos if}
+ * (sin <^ (1  A;* sin* 0)  A:* sin* ^ cos 0)}
= A:* sin* </> cos {(1  A;* sin* 0)  sin* (1  A;* sin* 0)}
+ k sin* Vl  A:* sin* ^ (1  A;* an* 0)
= A* sin* cos* + A? sin* ^ (1  A:* sin* 0)*,
and the denominator becomes
A;* sin* ^ cos* <^ — (1 — A;* sin* 0) sin* <f)
=  A/* sin* </>,
if A/* = lA:*.
Hence we have for the coordinates of the point of the caustic,
[A/«a? =  A:* cos* <^  A; (1  A?* sin* <^)*,
y= A?* sin* ^ ;
and eliminating 0, we obtain for the equation of the caustic,
A;'«ar = A^{l.A;*y*}*A;{lA;V}*;
or writing  instead of k, we find
(i/.*)^=(i/iV)*+/^(i/*"*y¥
for the equation of the caustic by refraction of the circle, for parallel raya The
equation was first obtained by St Laurent.
XIII.
The discussion of the preceding equation presents considerable interest. In the
first place to obtain the rational form write
a =(i/iO^, )8 = (i/.V)^7=/^(i/**y¥;
this gives
a*  20? (/S" + 7*) + (/S*  7')' = 0,
and we have
/3« = 1  3/i V + 3/tV  yity,
rf ^fi* 3/t%* + 3/t V  y»,
and consequently
/3«  y = (I  /I') {1  3/iV + (!+/*') y'i
44—2
348 A MEMOIR UPON CAUSTICS. [l45
Hence dividing out by the factor (1 — fi^y, the equation becomes
(l/Lt«)«a;*2(l+;i«6AtV+3/^*(l+/**)y*(l+/^')j^)2a?" + (l3AtV + ^^
or reducing and arranging,
+ (12/tVh9AtV)y^(6/^*(l+/^*)a^ + 6At* + 6At*(l+At»)y»)y* = 0,
which is of the form
A + 3/Lt*5y*  QfjfiCy* = ;
and the rationalized equation is
4* + 27 fi*B'i/'  21 6/i«(?y* + Bi^fi^ABCy* = 0,
where the values of A, B, C may be written
4 = (^ + y»){(l/.«)»^ + (H/i»)»y»}2(l+/.»)(a;»y») + l,
5 = 4«r« + 3y«,
the caustic is therefore a curve of the 12th order.
To find where the axis of x meets the curve, we have
y=0, 4o' = 0,
where
= {(!,.)» a;l}{(l+/i)»^l},
i.e.
fyO. ^
X— ± = , ir= +
1 " fi' 1 4/a'
or there are in all four points, each of them a point of triple intersection.
To find where the line oo meets the curve, we have
where
i.e.
^ 1 + /i» .
145] A MEMOIR UPON CAUSTICS. 349
or the curve meets the line oo in four points, each of them a point of triple
intersection: two of these points are the circular points at oo.
To find where the circle as* + y* = 1 meets the curve, this gives a;* = 1 — y*, and
thence
5 = 4y»,
and the equation becomes
{/A« (/*'  4) + 4 (1 + 2/i«) y«}« + 27/A* (4  y'/ y»  216 (/A« + 2)» y*
+ 54At»(Ai« + 2)y»(4y»){/Lt«(Ai«4) + 4(l+2At»)y»}=0,
which is only of the eighth order; it follows that each of the circular points at oo
(which have been already shown to be points upon the curve) are quadruple points
of intersection of the curve and circle. The equation of the eighth order reduces
itself to
(y»/iy{27Aiy + (M'4)»}=0;
the values of x corresponding to the roots y = + /a are obtained without difficulty,
and those corresponding to the other roots are at once found by means of the
identical equation
(/Lt«4)» + 27/i* + (l/i»)(^« + 8)» = 0;
we thus obtain for the coordinates of the points of intersection of the curve with
the circle a;* + y' = l, the values
a? = ± Vl  ^»
foo, a?=±V:
(a?=±ty, (y=±/f,
x= ± = t,
^ (m'  4)* .
each of the points of the first system being a quadruple point of intersection, each
of the points of the second system a triple point of intersection, and ectch of the
pomts of the third system a single point of intersection.
1 1
Next, to find where the circle a;*4y*=r— meets the curve; writing a;* = — — y»,
I* fit
we obtain for y an equation of the eighth order, which after all reductions is
(y*  ^)* {27/ty + (1  v/} = 0,
350
A MEMOIR UPON CAUSTICS.
[145
and we have for the coordinates of the points of intersection,
(
{:=
00
a? = ± iy,
X
1
(1 + 8m')
«= +
y =
^ (1  V)* •
?= ».
each of the points of the first system being a quadruple point of intersection, each
of the points of the second system a triple point of intersection, and each of the
points of the third systemi a single point of intersection.
The points of intersection with the axes of a?, and the points of triple inter
section with the circles aj" + y'=l and a;*fy« = — , are all of them cuspidal points;
the two circular points at oo are, I think, triple points, and the other two points of
intersection with the line oo, cuspidal points, but I have not verified this: assuming
that it is so,' there will be a reduction 54 accounted for in the class of the curve,
but the curve is, in fact, as will be shown in the sequel, of the class 6 ; there is
consequently a reduction 72 to be accounted for by other singularities of the curve.
XIV.
It is obvious from the preceding formulae that the caustic stands to the circle^^
radius , in a relation similar to that in which it stands to the circle, radius 1, Le.
to the refiracting circle. In fact, the very same caustic would have been obtained if^^
the circle radius — had been taken for the refracting circle, the index of refraction
being ~ instead of ^. This may be shown very simply by means of the irrational
form of the equation as follows.
The equation of the caustic by refraction of the circle, radius 1, index of refraction
/ir, is as we have seen
(i/.>)fl:=(i/iW+M(i/*»y¥;
hence the equation of the caustic by refiuction of the circle radius o\ index of
refiwjtion /i', is
(..^,i.{...(i)'}'...{i..()y,
145]
A MEMOIB UPON CAUSTICS.
351
or, what is the same thing,
which becomes identical with the equation of the firstmentioned caustic if /a' = c' = — .
Hence taking c instead of 1 as the radius of the first circle, we find,
Theorem. The caustic by refiaction for parallel ra}rs of a circle, radius c, index
of refraction /i, is the same curve as the caustic by refraction for parallel rays of a
concentric circle, radius  , index of refraction  .
XV.
We may consequently in tracing the caustic confine our attention to the case in
which the index of refiuction is greater than imity. The circle, radius , will in this
case be within the refracting circle, and it is easy to see that if from the extremity
of the diameter of the refiucting circle perpendicular to the direction of the incident
rays, tangents are drawn to the circle, radius  , the points of contact are the points
of triple intersection of the caustic with the lastmentioned circle, and these points
of intersection being, as already observed, cusps, the tangents in question are the
tangents to the caustic at these cusps. The points of intersection with the axis of
X are also cusps of the caustic, the tangents at these cusps coinciding with the axis
of x: two of the lastmentioned cusps, viz. those whose distances from the centre are
1 . . . . . c
± , lie within the circle, radius  , the other two of the same four cusps, viz.
M + 1 M
those whose distances from the centre are ±
Ml
, lie without the circle, radius
 ; the lastmentioned two cusps lie without the refracting circle, when /a < 2, upon
this circle, when /a = 2, and within it and therefore between the two circles, when
/A>2. The caustic is therefore of the forms in the annexed figures 3, 4, 5, in each
Fig. 3.
Fig. 4.
Fig. 5.
T"^.
**..
352 A MEMOm UPON CAUSTICS. [145
of which the outer circle is the refracting circle, and fi ib > 1, but the three figures
correspond respectively to the cases ft < 2, ft = 2 and /k > 2. The same three figures
will represent the different forms of the caustic when the inner circle is the refracting
circle and fi is < 1, the three figures then respectively corresponding to the cases
A^ > i> /^ = i> ^^ /* < i
XVI.
To find the tangential equation, I retain k instead of its value ; the equation
of the refructed ray then is
a? (A; cos  Vl  ^in« <^) + y (k sin <^ + cot ^ Vl  A:» sin* <t>)  Jfc = 0,
and representing this by
Zar+FyA;=0,
we have
Z = A;cos0VlA;»sin»<^,
F = A: sin ^ 4 cot ^ Vl — A:* sin* 0,
equations which give
X cos + Fsin ^ = A?,
z«+r* ^
and consequently
sin* <l> '
. . 1
sm = ,—
, Vz*+y*i
cos 6 = — .— — 
^ VZ* + F*
^ )
and we have
which gives
ZVZ*+ F*l + Fifc VZ*+ F* = 0,
(Z«+F*)(Z*lJfc*) = 2A;FVZ*+F*;
or, dividing out by the factor VZ* + F*, the equation becomes
\/Z*+F*(Z* 1Jfc*) =  2A;F,
from which
(Z*+F*)(Z*lJfc*)*4ifc»F* = 0;
or reducing and arranging, we obtain
Z*(Z*lAj*)*+F*(Z + l+A;)(Z + lA:)(Zl+ifc)(Zli) =
for the tangential equation of the caustic by refraction of a circle for parallel raj^s.
The caustic is therefore of the class 6.
145] A MEMOIR UPON CAUSTICS. 353
XVII.
Suppose next that rays proceeding from a point are reflected at a circle.
A very elegant solution of the problem is given by Lagrange in the Mim. de
Turin; the investigation, as given by Mr P. Smith in a note in the Cambridge and
Dublin Mathematical Journal, t. ii [1847] p. 237, is as follows:
Let B be the radiant point, RBP an incident ray, and PS a reflected ray; CA
a fixed radius; ACP = a, ACB = €, reciprocal of CB = Cy reciprocal of CP = a, The
equations of the incident and reflected ray, where w =  , may be written
u=^Asin0 + B cos ; incident ray,
t* = i4sin(2a — ^) + 5cos(2a — ^); reflected ray,
the conditions for determining A and B being
a = ^ sin a 4 5 cos a,
c = A sin e + B cos e,
whence
. _ a cos €  c cos a p _ c sin a — a si n e
sm (a — e) sin (a — e)
Substituting these values, the equation of the reflected ray becomes
a sin (2a — ^ — e) = M sin (a — 6) 4 c sin (a — ^),
firom which and its differential with respect to the arbitrary parameter ce, the equation
of the caustic, or envelope of the reflected rays, will be found by eliminating a.
In this, a being the only quantity treated as variable in the differentiation, let
2a^e = 20,
therefore
a iL
45
354 A MEMOIR UPON CAUSTICS. [l45
and the equation becomes
a sin 20 = w sin {<^ + J (^  €)} + c sin [<f>^{0 e)}.
Make
also
p __ (<f + c) cos ii^ — e)
^ 2a
] 1
then the equation becomes
with the condition
Hence
cos ' ^ sin (^ '
P^ + Qy = i.
•»"'+y"*=i.
multiplying by x and y, and adding, we find X = 1 ; therefore
Hence
or restoring the values of P and Q,
{(16 + c) cos i (^  e)}* + {(u  c) sin ^{0 e)}* = 1,
the equation of the caustic.
XVIII.
But the equation of the caustic for rays proceeding from a point and reflected
at a circle may be obtained by a different method, as follows:
Take the centre of the circle for origin ; let c be the radius of the circle, fl, ^
the coordinates of the radiant point, a, the coordinates of the point of incidence,
X, y the coordinates of a point in the reflected ray. Then we have from the equation
of the circle o^\^'=^&, and the equation of the reflected ray is by the general
formula,
(ft'x  a/8) (cm? + /8y  c») + (ya  a?/8) (aa + 6/8  c*) = ;
145] A MEMOIR UPON CAUSTICS. 355
or arraDging the terms in a diflferent order,
(6a? + ay)(a»)8') + 2(6ycur)a^c^(6+y)a + c»(a + a?))8 = 0;
and writing herein a = c cos 0, /8 = c sin ^, the equation becomes
(bx + ay) cos 2^ + {hy — olx) sin 2^ — (6 + y) c cos ^ + (a + a:) c sin ^ = 0,
where d is a variable parameter.
Now in general to find the envelope of
A cos 2^ + J5 sin 2^+ C cos ^ + 2) sin ^ + ^ = 0,
we may put e^==z, which gives the equation
and equate the discriminant to zero : this gives
(4/)»  27 ( 8J)» = 0,
where
8J = 4((?2>»)+25C'2){8(^'' + £«) + ((? + 2>»)}i^+^^,
and consequently
 27 [A ((? Z)«) + 2BGD  (8 {A^^B") + (0+ Z>«)) ^^ + ^J5?}« = 0;
«nd substituting for Ay B, C, D, E their values, we find
f 4 (a« + 6») (aj" + y*)  c« ((a? + a)« + (y + 6)»)}' 27 (bxayy (a^ + f a^^¥y= 0,
ibr the equation of the caustic in the case of rays proceeding from a point and
xeflected at a circle: the equation was first obtained by St Laurent.
It will be convenient to consider the axis of ^ as passing through the radiant point ;
this gives 6 = 0; and if we assume also c = l, the equation of the caustic becomes
{(4a«  1) (aj" + y«)  2gw?  a«}»  27ay (ic» + y» a»)» = 0.
XIX.
Reverting to the equation of the reflected ray, and putting, as before, c = 1, 6 = 0,
tihis becomes
/ « /I . 1 \ a cos 2^ — cos ^ ^
(2acos^ + l)a?+ ^— ^ t/ + a = 0;
^ ^ sm r "^
>
differentiating with respect to 0, we have
(2asm^)ar + . ,^ y = 0;
45—2
356
and from these equations
A MEMOIR UPON CAUSTICS.
[■
_ tt^cos (1 h28 i n«g)~a
^ " 1 ~3a cos 2dT2a«
y =
2a« sin» ^
1 3a cos 2^+ 2a*'
which give the coordinates of a point of the caustic in terms of the angle wl
determines the position of the point of incidence. The values in question satisfy,
they should do, the equation
{(4a»  1) (aj" + y«)  2aa?  a»}»  27ay (ic» + y»  a«)« = 0;
we have, in fact,
4a* (cos 0—af
^+y'
tt» =
(4a« 1) (ar» + y«)  2aa;  a' =
(1  3a cos 2^ + 2a«)» '
12aHcos^ay
(l3acos2^+2a«)>'
from which it is easy to derive the equation in question.
XX.
If we represent the equation of the reflected ray by
Xx^Yy + a 0,
then we have
X = 2acos^ + l,
a cos 20 — cos
F=
sin
and thence
(Z  1)>  4a« =  4a« sin« ^,
X'+Y* = .\ >, (1  2a cos + a»),
sm^ 0^
X + a^ =l2aco80 + a\
and consequently
(X*+Y') {(Z 1)» 4a»l + 4a»X + 4a* = 0,
or, what is the same thing,
{Z(Zl)2a«}«+F='{(Zl)«4a'} = 0,
which may be considered as the tangential equation of the caustic by refle:
circle; or if we consider X, F as the coordinates of a point, then the equ;
be considered as that of the polar of the caustic. The polar is therefore a cv
fourth order, having two double points defined by the equations X(Z — 1) — 2a*
145] A MEMOIR UPON CAUSTICS. 357
and a third double point at infinity on the axis of F, i.e. three double points in all ;
the number of cusps is therefore 0, and there are consequently 4 double tangents and
6 inflections, and the curve is of the class 6. And as F is given as an explicit
function of X, there is of course no difficulty in tracing the curve. We thus see
that the caustic by reflexion of a circle is a curve of the order 6, and has 4 double
points and 6 cusps (the circular points at infinity are each of them a cusp, so that
the number of cusps at a finite distance is 4) : this coincides with the conclusions
which will be presently obtained by considering the equation of the caustic.
XXI.
The equation of the caustic by reflexion of a circle ia
{(4a«  1) (a:» + y«)  2cw?  a»}»  27ay (ar» + y*  a^y = 0.
Suppose first that y = 0, we have
{(4a«  1) a:*  2cw?  a*}« = 0,
— a __ a
^^' ^'isr+i' ^"2^r
or the curve meets the axis of a? in two points, each of which is a triple point of
intersection.
Write next aJ" + y'=a^ this gives
{(4a«  1) a«  2aa?  a«}» = 0,
and consequently
a? =  a (1  2a«),
y = ± 2a» Vi  a«,
or the curve meets the circle a;* + y*— a^ = in two points, each of which is a triple
point of intersection.
To find the nature of the infinite branches, we may write, retaining only the terms
of the degrees six and five,
(4a« l)'(a;» + y»)» 6 (4a«  l)«a (a;» + y*)» a? 27 ay (a;» + y*)^ = ;
and rejecting the factor (oc^ + jf^y, this gives
(4a»l)»«»+{(4a«l)»27a«}y»6(4a»l)»aa: = 0;
or reducing,
(4a«l)»a;»(la')(8a» + l)»y»6(4a*l)»aa?=0;
and it follows that there are two asymptotes, the equations of which are
_(4^_l)*_ L 3^ 1
VT^*(8a' + l)l 4a«ir
358 A MEMOIR UPON CAUSTICS. [l45
Represent for a moment the equation of one of the asymptotes by y = A{x — o),
then the perpendicular from the origin or centre of the reflecting circle is iias Vl \A^,
and
3a V4a*  1
Aa =
Vla«(l+8aO
I , ^,^ (la')(l+Sa«)' + (4a«l) »^ 27a'
Vl ^A^ =
(1  a«) (1 + 8a»)« (1  a«) (1 + 8a»)^ '
3V3a
Vl  a« (1 + 8a«)
and the perpendicular is ^V4r^»_i, which is less than a if only a«<l, i.e. in every
case in which the asymptote is real.
The tangents parallel and perpendicular to the axis of x are most readily obtained
from the equation of the reflected ray, viz.
/ « /» 1 \ a cos 2^ — cos ^ ^
{— 2a cos ^+1) a; H : — ^ v + a = 0:
the coefficient of x (if the equation is first multiplied by sin^) vanishes if sin = 0,
1 V4a' — 1
which gives the axis of a?, or if cos = — , which gives y = ± — ^ , for the tangents
parallel to the axis of x.
The coefficient of y vanishes if a cos 20 — cos 5 = 0; this gives
^ 1 + V8a« + 1 . ^ 1 , , , ^ /s5 — ?x
cosg=  ^ , 8ine=^ (4a»l T v8a«+l),
4a oar
and the tangents perpendicular to the axis of x are thus given by
2a
1 + V8a« + 1 '
these tangents are in fact double tangents of the caustia In order that the point of
contact may be real, it is necessary that sin 0, cos should be real ; this will be the
case for both values of the ambiguous sign if o > or = 1, but only for the upper
value if a < 1.
It has just been shown that for the tangents parallel to the axis of x, we have
V4a«1
y = ±
2a
V4a' — 1
the values of y being real for a > ^ : it may be noticed that the value y = —
145] A MEMOIR UPON CAUSTICS. 359
is greater, equal, or less than or to y = 2a' Vl — a', according as a > = or < 7= 5 ^^^
depends on the identity (4a»  1)  16a« (1  a«) = (2a»  1)« (2a» +1).
To find the points of intersection with the reflecting circle, ic* + y' — 1 = 0, we have
(3a«  1  2cw:)» 27a» (1  a;»)(l  a«>» = ;
or, reducing,
8aV + ( 27a* + 18a» 15) a'ic* + (54a*  36a» + 6) oo? + ( 27a* + 18a«+ 1) = 0,
Le. (aa?l)»(8aa?27a*+18a» + l) = 0.
The factor (ax^iy equated to zero shows that the caustic touches the circle in
the points a? =  , y = ± a/ 1 ^ , i e. in the points in which the circle is met by the
polar of the radiant point, and which are real or ima^nary according as a > or < 1.
The other &ctor gives
27a*  18a«  1
x =
8a
Putting this value equal to ± 1, the resulting equation is (a + l)(27a' + 9a + l) = 0, and
it follows that x will be in absolute magnitude greater or less than 1, Le. the points
in question will be imaginary or real, according as a>l or a<l.
It is easy to see that the curve passes through the circular points at infinity,
and that these points are cusps on the curve ; the two points of intersection with the
axis of X are cusps (the axis of x being the tangent), and the two points of inter
section with the circle a^ + y' — a' = are also cusps, the tangent at each of the cusps
coinciding with the tangent of the circle ; there are consequently in all 6 cusps.
XXII.
To investigate the position of the double points we may proceed as follows: write
for shortness P = (4a*l)(a:* + y') 2aa?a*, Q = ar/S, S^xkfa}) the equation of the
caustic is
P»27Q» = 0;
hence, at a double point,
one of which equations may be replaced by
dP dQ^dP dQ^^
dx dy dy dx '
360 A MEMOIR UPON CAUSTICS. [145
Now —
^£ = 2{(^'l)xa], ^ = 2(4a'l)y.
^=2aay. ^ = o(^ + 3y'a») = a(S+2y«);
substituting these values in the last preceding equation, we find
~(4a»l)"y "Sh2y«*
or, reducing,
and using this to simplify the equation
we have
^s'««S».
P'^lSay8.2axy = 0,
i.e. ^ ^ — 9cuF/S> = 0,
and therefore
Multiplying by P and writing for P* its value 27a^y'S^, we have
Px = 3ay«,
and thence
whence
«y* = 5» ^ = S or 2^=?!
a!» • J • S „i'
or "a'
and substituting in the equation
* 4a'U Sj'
we find
or, rationalising,
4(M!»  {(4o'  1) «  o)' = 0,
145] A MEMOIR UPON CAUSTICS. 361
or, what is the same thing,
(4flw?l)(a?a(a + iVl a«)») (a:a(aiVl a»)«) = 0.
The factor 4tax — l equated to zero gives ^=7 from which y may be found, but the
resulting point is not a double point; the other factors give each of them double
points, and if we write
we find
^ 2aH'(a + iVla')*
values which, in fact, belong to one of the four double points. It is easy to see that
the points in question are always imaginary.
It may be noticed, by way of verification, that the preceding values of x, y give
(4a«  l)(«" + y')  2flw:  a» = ^^^^^ (1  4a» 4ai VTIT^),
x' + y»a' = jjg^, (3a + i Vl  a« ),
f = ^t."^, ( 1 + 14a»  16a« + 2a (3  8a») i Vl  a*) ;
and if the quantities within ( ) on the righthand side are represented by A, B, C, then
i (a + iVrr^').
whence we have identically,
B
f)
_ = _(a4zVla»)>,
(^)' = 5'  ^'=^^'
by means of which it appears that the values of x, y satisfy, as they should do, the
equation of the caustic ; and by forming the expressions for (4a'— l)a; — a and a^ + S^*— a',
it might be shown, d posteriori^ that the point in question was a double point.
XXIII.
The equation
{(4a« 1) (a;* + y«)  2aa?  a«}»  27ay (a^ + y'  a«)» =
l)ecome8 when a = 1 (L e. when the radiant point is in the circumference),
{3y> + (a?  1) (3a?+ 1)}»  27y« (y» + «»  1)» = ;
it is easy to see that this divides by (a?— 1)*; and throwing out this factor, we have
for the caustic the equation of the fourth order,
27 j^ + 18y« (3ar» _ 1) + (a?  1) (3a; + 1)» = 0.
A /•
362
A MEMOIR UPON CAUSTICS.
[145
XXIV.
The equation
{(4a«  1) (a^ + jf")  200?  a»}»  27ay (ic* + y»  a«)» =
becomes when a = x (i. e. in the case of parallel rays),
(4ic» + 4y«l)»27y» = 0.
which may also be written
64>af + 48a?* (4y«  1) + 12a?»(4y»  1)» + (8y« + 1)» (y*  1) = 0.
XXV.
It is now easy to trace the curve. Beginning with the case a = oo , the curve lies
wholly within the reflecting circle, which it touches at two points; the line joining
the points of contact, being in fact the axis of y, divides the curve into two equal
portions ; the curve has in the present, as in every other case (except one limiting
case), two cusps on the axis of x (see fig. 6). Next, if a be positive and > 1, the
general form of the curve is the same as before, only the line joining the points of
contact with the reflecting circle divides the curve into unequal portions, that in the
Fig. 7. a>l.
Fig.
6.
a — CD.
,•
i^S^^
1 ^^^."^^
yy
/ /
j \\
/ V
[ J \
i f
. ...^....f.
\ 1
1 \ '
\ I
I J /
\ \
• / /
*» V
1 y ^
''C^
Sw
neighbourhood of the radiant point being the smaller of the two portions (see fig. 7).
When a = 1, the two points of contact with the reflecting circle unite together at the
radiant point ; the curve throws off, as it were, the two coincident lines a? = 1, and the
order is reduced fiom 6 to 4. The curve has the form fig. 8, with only a single cusp
on the axis of x. If a be further diminished, a < 1 > = , the curve takes the form
v2
shown by fig. 9, with two infinite branches, one of them having simply a cusp on
the axis of x, the other having a cusp on the axis of m^ and a pair of cusps at its
intersection with the circle through the radiant point; there are two asymptotes equally
inclined to the axis of x. In the case a — j=^^ the form of the curve is nearly the
same as before, only the cusps upon the circle through the radiant point lie on the
axis of y (see fig. 10). The case a<~>\ is shown, fig, 11. For = ^, the two
145]
A MEMOIR UPON CAUSTICS.
363
asymptotes coincide with the axis of a?; one of the branches of the curve has wholly
disappeared, and the form of the other is modified by the coincidence of the asymptotes
Fig. 8. a=l.
Fig. 13. a = l
\
\ \
Fig. 11,
^ 1
v A
.».,—>..
i X4
Fig. 10. a
V2'
Fig. 12. a = J.
\
• — i —
with the axis oi x\ it has in fact acquired a cusp at infinity on the axis of x (see
fig. 12). When a<\, the curve consists of a single finite branch, with two cusps on
the axis of x^ and two cusps at the points of intersection with the circle through
the radiant point; one of the lastmentioned cusps will be outside the reflecting circle
as long as o>J; fig. 13 represents the case a=^, for which this cusp is upon the
reflecting circle. For a<J, the curve lies wholly within the reflecting circle, one of
the cusps upon the axis of x being always within, and the other always without the
circle through the radiant point, aod as a approaches the curve becomes smaller
and smaller, and ultimately disappears in a point. The case a negative is obviously
included in the preceding one.
Several of the preceding results relating to the caustic by reflexion of a circle
were obtained, and the curve is traced in a memoir by the Rev. Hamnet Holditch,
(^Tierly MathemcOical Journal, t i. [1857, pp. 93 — 111].
46—2
364
A MEMOIR UPON CAUSTICS.
[145
XXVI.
Suppose next that rays proceeding from a point are refraxiied at a circle. Talie
the centre of the circle as origin, let the radius be c, and take ^, i; as the coordi
nates of the radiant point, a, /9 the coordinates of the point of incidence, x, y the
coordinates of a point in the refracted ray : then the general equation
qG VQON +n*QO '7qON =0
becomes, taking the centre of the circle as the point N on the normal, or writinj^
a=b, 6 = 0,
or putting a*(y3' = C, and expanding,
a' {2 (i,»ar  /iVf )}
+ a«/8 { 4 (^x  /i'an/f ) + 2(ryy ^*fn)\
+ a/S* { 4 (fi,y  ii}xyf,) + 2 (pa:  /iVf)}
o« {(a!» + y» + c»)i?» M* (?+'?* + C)!^}
+ 2a^8 ((a^ + y» + c») fv  /*« (f» H 1;» + c*) ay}
/3» {(«»ly' + c»)p/t«(f» + i7» + c»)a?}
= 0,
which may be represented by
4a« + P««/8 Ca/g' + D/S'l .P«» + ^0/9 + H0' = O.
Now a* + /8* = c*, and we may write
The equation thus becomes
l<J)'!«(l)('i)H'i)'»
or expanding,
+ (FGiH) ^
+ (3^£t + C + 3i)i) «
+ *(F+H)
^=0.
+ (3il + £» + C  3IH") 
l(.P((?i5)
c
1
+ (4 + 5i  C + IH) i
145] A MEMOIR UPON CAUSTICS. 365
n which z may be considered as the variable parameter ; hence the equation of the
austic may be obtained by equating to zero the discriminant of the above function
>f z\ but the discriminant of a sextic function has not yet been calculated. The
equation would be of the order 20, and it appears from the result previously obtained
or parallel rays, that the equation must be of the order 12 at the least ; it is, I think,
)robable that there is not any reduction of order in the general case. It is however
)racticable, as will presently be seen, to obtain the tangential equation of the caustic
)y refraction, and the curve is thus shown to be only of the class 6.
XXVII.
Suppose that rays proceeding from a point are refracted at a circle, and let it be
•equired to find the equation of the secondary caustic: take the centre of the circle as
>rigin, let c be the radius, f , i; the coordinates of the radiant point, a, fi the coordinates
)f a point upon the circle, /la the index of refraction; the secondary caustic will be
ihe envelope of the circle,
where a, /8 are variable parameters connected by the equation a' + zS' — c* = 0; the
liquation of the circle may be written in the form
But in general the envelope of 4a + £/8 + C = 0, where a, /8 are connected by the
equation a' + ^S* — c'ssO, is & {A} + JP) — (? = 0, and hence in the present case the equation
of the envelope is
which may also be written
{/it« (a^ + y»  c*)  (p + 17»  c«)}« = 4c«/it« {(a;  f )« + (y  17)»}.
If the axis of x be taken through the radiant point, then 97 =0, and writing also
{ = a, the equation becomes
or taking the square root of each side,
{/A« (a;* + y»  c»)  a« + c"} = 2cm V(a?a)» + y« ;
1 . / IV
whence multiplying by 1 — 5 and adding on each side c* ( /^ — ) + (^ — a)^ + y', we have
/**(«^,y + yj = V(^a)« + y« + c(MJ)}',
or
'^V (^~5) +y'='^(^«)"+y'+^(^^)'
which shows that the secondary caustic is the Oval of Descartes, or as it will be con
venient to call it, the Cartesian.
366 A MEMOIB UPON CAUSTICS. [145
It is proper to remark, that the Cartesian consists in general of two ovals, one
of which is the orthogonal trajectory of the refracted rays, the other the orthogonal
trajectory of the false refracted raya In the case of reflexion, the secondary caustic
is a Cartesian having a double point ; this may be either a conjugate point, or a real
double point arising from the union and intersection of the two ovals ; the same
secondary caustic may arise also from refraction, as will be presently shown.
XXVIII.
Reverting to the original form of the equation of the secondary caustic, multipljring
by — J (1 i] and adding on each side ~fl ij +~, {(^ — a)* + y'j, the equation
becomes
or extracting the square root,
Combining this with the former result, we see that the equation may be expressed
indifferently in any one of the four forms,
^/(f)'^'=^;:^/(^J)>^'^
("  ^) \/('  9"  '• ^ ( <■ + ^ \/('  J.)'  ^  e  t) ^'<^^^>^ = «■
It follows, that if we write successively
a'= a, c' = c f /*' = /* (1)
a =  , c =  , M =  (a)
a fi a ^ ^
a =  2 , c = , /A =  (p)
a =a , c =  , M =  (7)
/x c
a=— , C =C y a = — (0)
145] A MEMOIR UPON CAUSTICS. 367
or what is the same thing,
(1)
(a)
(0)
(7)
or what is again the same thing,
a==a' ,
c = c',
t*=l*'
a'
a'
c= ,
A*
a'
of
c'
1
a =a' ,
a'
a'
a =>,
a
c = c' ,
ey
a
a
c'
c'
a
J thing,
o' = a.
a' a
a
c'» a
a' /*»'
a'
a a
a'
;.'. = «
a' = a ,
c'' a
a
a
a' a
a
/» a
(8)
(«).
(1)
(«)
(/8)
(7)
(S)
(0.
we have in each case identically the same secondary caustic, and therefore also
identically the same caustic ; in other words, the same caustic is produced by six
different systems of a radiant point and refracting circle. It is proper to remark that if
we represent the six systems of equations by (a', c', fi!) = (a, c, /i), {a\ c', /i') = a (a, c, /a),
&c., then, a, )3, 7, S, 6 will be functional symbols satisfying the conditions
a = /S" = &y = eS =76,
/9 = a* = 78 = Sc = €7,
<y = Sa = a€ = 6/8 = ^S,
5 = ea = 07 = 7/3 =/8€,
6 =:7a = aS=: S/8 =)87.
368 A MEMOIR UPON CAUSTICS. [145
XXIX.
The preceding formulae, which were first given by me in the Philosophical Magdzine,
December 1853, [124] include as particular cases a preceding theorem with respect
to the caustic by refraction of parallel rays, and also two theorems of St Laurent,
Oergonne, t. xviii., [1827, pp. 1 — 19] viz. if we suppose first that a=^Cy Le. that the
radiant point is in the circumference of the refracting circle, then the system (a) shows
that the same caustic would be obtained by writing c,  , 1 (or what is the same
thing — 1) in the place of c, c, /x, and we have
Theorem. The caustic by refraction for a circle when the radiant point is in the
circumference is also the caustic by reflexion for the same radiant point, and for a
reflecting circle concentric with the refracting circle, but having its radius equal to the
quotient of the radius of the refracting circle by the index of refraction.
Next, if we write a = CfjLj then the refracted rays all of them pass through a point
which is a double point of the secondary caustic, the entire curve being in this case
the orthogonal trajectory, not of the refracted rays, but of the false refracted rays; the
formula (S) shows that the same caustic is obtained by writing , c, 1 (or what is
Cv
the same thing — 1) in the place of a, c, /i f =  j , and we have
Theorem. The caustic by refraction for a circle when the distance of the radiant
point from the centre is to the radius of the circle in the ratio of the index of
refraction to unity, is also the caustic by reflexion for the same circle considered as
a reflecting circle, and for a radiant point the image of the former radiant point
XXX.
The curve is most easily traced by means of the preceding construction ; thus if
we take the radiant point outside the refracting circle, and consider fi as varying fix)m
a small to a large value (positive or negative values of /i give the same curve), we
see that when fi is small the curve consists of two ovals, one of them within and
the other without the refracting circle (see fig. 14). As /i increases the exterior oval
continually increases, but undergoes modifications in its form; the interior oval in the
first instance diminishes until we arrive at a curve, in which the interior oval is reduced
to a conjugate point (see fig. 15); then as fi continues to increase the interior oval
reappears (see fig. 16), and at last connects itself with the exterior oval, so as to
form a curve with a double point (see fig. 17); and as /i increases still further the
145]
A MEMOIR UPON CAUSTICS.
369
curve again breaks up into an exterior and an interior oval (see fig. 18) ; and thence
forward as fjL goes on increasing consists always of two ovals; the shape of the exterior
oval is best perceived fh)m the figures. An examination of the figures will also show
how the same curves may originate from a different refracting circle and radiant point.
Fig. 14.
Fig. 17.
Fig. 15.
Fig. 18.
0. II.
47
370 A MEMOIB UPON CAUSTICS. [l45
XXXI.
The theorem, " If a variable circle have its centre upon a circle S, and its radius
proportional to the tangential distance of the centre from a circle (7, the envelope is
a Cartesian,"
is at once deducible from the theorem —
'' If a variable circle have its centre upon a circle S and its radius proportional
to the distance of the centre from a point C\ the locus is a Cartesian,'*
which last theorem was in eflfect given in discussing the theory of the secondary
caustic. In fact, the hx^us of a point P such that its tangential distances from the
circles C, C" are in a constant ratio, is a circle S, Conversely, if there be a circle C,
and the locus of P be a circle S, then the circle C" may be found such that the
tangential distances of P from the two circles are in a constant ratio, and the circle
C may be taken to be a point, i.e. if there be a circle C and the locus of P be
a circle S, then a point C may be found such that the tangential distance of F
from the circle C is in a constant ratio to the distance from the point C\
Hence treating P as the centre of the variable circle, it is clear that the variable
circle is determined in ihe two cases by equivalent constructions, and the envelope is
therefore the same in both cases.
XXXII.
The equatiiJii of the secondary caustic developed and reduced is
^* (x'' + yy  2/x* (a** \{fi^+l) c^) (a^ + y^) + 8cV(W 4 a*  2aV (/x^ + 1 ) + (/x^  1 ) c* = (
or, what is the same thing,
which niav also be written
which is of the form
{x" 4 3/'  ay h 16.4 (./■  m) = :
and the values of the coefficients are
m
^,^ + 0+^)a\
145]
A MEMOIR UPON CAUSTICS.
371
The equation just obtained should, I think, be taken as the standard form of the
equation of the Cartesian, and the form of the equation shows that the Cartesian may
be defined as the locus of a point, such that the fourth power of its tangential
distance from a given circle is in a constant ratio to its distance from a given line.
XXXIII.
The Cartesian is a curve of the fourth order, symmetrical about a certain line
which it intersects in four arbitrary points, and these points determine the curve.
Taking the line in question (which may be called the axis) as the axis of x^ and a
line at right angles to it as the axis of y, let a, 6, c, d be the values of x corre
sponding to the points of intersection with the axis, then the equation of the curve is
#
+ (.T  a) (x b)(x c) (xd) = 0.
It is easy to see that the form of the equation is not altered by writing x\0 for x,
and a\0y b\0, c + O, d\0 for a, 6, c, d, we may therefore without loss of generality
put a\b \ c + d = 0, and the equation of the curve then becomes
3/* + y2 (2^ + aft + ac + ad h 6c h M + cd) + (a:  a) (a?  6) (x  c) (a;  d) = 0,
where
a\b\c\d = 0',
the cur\'e is in this case said to be referred to the centre as origin.
The lastmentioned equation may be written
(^ + y^y \(ab\ac + ad\bc\bd{cd){af^ + j/^) (abc h aid + acd f bed) x + abed
= 0,
or
[a^ \ y"" {^ {db +ac + ad\bc \bd \cd)]'
— (abc + ahd + acd h bed) x
ii
= 0,
or
( a^¥ + aV + a^'d^ + b^(^ + h'd? + c^d?
+ 2a26c + 2a26d h 2a^cd + 26«ac + Wad h 2b^ed
+ 2c«a6 + 2c2ad h 'Id'bd + id'ah h ^d'ac h 2d'6c
^ +2a6cd
observing that
a^be + a'hd H a^'ed H b^ae H 6 W h Ifed
+ c^'ab + e^ad + e^'bd 4 d^ab + d^ac h d?he
= a6c (a + 6 + c) H abd {a\b\d)\ acd {a \ e {• d) \bed{b ^ c ■\ d)
372 A MEMOIR UPON CAUSTICS. [l45
the equation becomes
j^i" + y' + i (a6 + ac + od + 6c + M + cd)}«
— (abc + abd + acd + bed) x
 \ (a«6« + a^\d^d^ + 6«c« + 6«d* + c»d* 6a6cd) = 0,
which is of the fonn
(j?* + y«  a)» + 164 (a?  7^1) = 0,
and, as already remarked, signifies that the fourth power of the tangential distance
a point in the curve from a given circle, is proportional to the distance of the sai
point from a given line. The circle in (juestion (which may be called the dirige^^^
circle) has for its equation
^ + y' + i («^ + «c + ad + 6c + 6d + cd) = ;
the line in question, which may be called the directrix, has for its equation
d}V + a^c^ + a«d« + 6'c^ + h^d? + Cd*  6a6cd ^
/p j_ — . f I «
4 (oic + abd + ojcd + hcd) '
the multiplier of the distance from the directrix is
abc + abd + acd + hcd.
It may be remarked that a, 6, c, d being real, the dirigent circle is real; the equation
may, in fact, be written
^ + y' = 4 [(a + &)' + (a + c)« + (a + d)« + (6 + c)» + (6 + d)» + (c + d)>].
XXXIV.
Considering the equation of the Cartesian under the form
(a?> + y'  (jLf + 164 (a:  m) = 0,
the centre of the dirigent circle a;* + y' — a = must be considered as a real point,
but a may be positive or negative, i.e. the radius may be either a real or a pure
imaginary distance: the coefficients A, m must be real, the directrix is therefore a real
line. The equation shows that for all points of the curve a? — m is always negative
or always positive, according as il is positive or negative, i.e. that the curve lies
wholly on one side of the directrix, viz. on the same side with the centre of the
dirigent circle if 4 is positive, but on the contrary side if A is negative. In the
former case the curve may be said to be an * inside * curve, in the latter an ' outside '
curve. If 7/1 = 0, or the directrix passes through the centre of the dirigent circle,
then the distinction between an inside curve and an outside curve no longer exists.
It is clear that the curve touches the directrix in the points of intersection of this
line and the dirigent circle, and that the points in question are the only points of
intersection of the curve with the directrix or the dirigent circle; hence if the
directrix and dirigent circle do not intersect, the curve does not meet either the
directrix or the dirigent circle.
145] A MEMOIR UPON CAUSTICS. 373
XXXV.
To discuss the equation
I write first y = 0, which gives
for the points of intersection with the axis of x. If this equation has equal roots,
there will be a double point on the axis of x, and it is important to find the
condition that this may be the case. The equation may be written in the form
(3, 0, a, 12A, 3a»48ilm$a?, 1)* = 0,
the condition for a part of equal roots is then at once seen to be
 (a«  l2Amy + (a»  18il77ia + 54^2)' = ;
or reducing and throwing out the factor ^', this is
27 A^ + 2m (8m« da) A a« (m«  a) = 0.
This equation will give two equal values for A if
m» (8m'  9ay + 27a' (m»  a) = 0,
an equation which reduces itself to
(4m'  Say = 0.
4m'
Hence, if 4wi' — 3a be negative, i.e. if a>^, the values of A will be imaginary,
»j
4m' . 4m'
but if 4m' — 3a be positive, or a <  . , the values of A will be real. If a = —^ ,
then there will be two equal values of A, which in fact corresponds to a cusp upon
the axis of x. Whenever the curve is real there will be at least two real points on
the axis of x; and when a<^, but not otherwise, then for properly selected values
of A there will be four real points on the axis of x.
Differentiating the equation of the curve, we have
{(af^ + fa)x + 4iA) dx + {a^tif' a) ydy = ;
and if in this equation we put dx = 0, we find y=0, or ar'fy' — a = 0, i.e. that the
points on the axis of x, and the points of intersection with the circle a^\y^ — a = 0,
are the only points at which the curve is perpendicular to the axis of x. To find
the points at which the curve is parallel to the axis of a;, we must write (ir = 0, this
gives
(a;' + y'  a) X + 4^ = 0,
374
A MEMOIR UPON CAUSTICS.
[145
and thence
ic* + y* — a = —
4^
X
and
^ + a^ (a?  m) = :
this equation will have three real roots if ^<^=, and only a single real root if
27
4mv
4m
A > ^= ; for J. = ^ , the equation in question will have a pair of equal roots. It
is easy to see that there is always a single real root of the equation which gives
rise to a real value of y, Le. to a real point upon the curve; but, when the equation
has three real roots, two of the roots may or may not give rise to real points upon
the curve.
XXXVI.
It is now easy to trace the curve. First, when m = 0, or the directrix
through the centre of the dirigent circle, the curve is here an oval bent in so as
to have double contact with the directrix, and lying on the one or the other side of
the directrix according to the sign of A, See fig. a.
Fig. a.
Fig. h.
..'
Fig. c.
Fig. d.
145] A MEMOIR UPON CAUSTICS. 375
Next, when the directrix does not pass through the centre of the dirigent circle,
it will be convenient to suppose always that m is positive, and to consider A as
passing first from to oo and then from to — oo , i. e. to consider first the
different inside curves, and then the different outside curves. Suppose a > —^ , the
o
inside curve is at first an oval, as in fig. 6, where (attending to one side only of
the axis) it will be noticed that there are three tangents parallel to the axis, viz.
one for the convexity of the oval, and two for the concavity. For A = ^ the two
tangents for the concavity come together, and give rise to a stationary tangent (Le. a
tangent at an inflection) parallel to the axis, and for ^ > ^ the two tangents for
the concavity disappear. The outside curve is an oval (of course on the opposite side
of, and) bent in so as to have double contact with the directrix.
Next, if a = —^ , the inside curve is at first an oval, as in fig. c, and there are,
as before, three tangents parallel to the axis : for A = ^= , the tangents for the con
cavity of the oval come to coincide with the axis, and are tangents at a cusp, and
for A > g^^ the cusp disappears, and there are not for the concavity of the oval any
tangents parallel to the axis. The outside curve is an oval as before, but smaller and
more compressed.
Next, a < ^ > m*, then the inside curve is at first an oval, as in fig. d, and
there are, as before, three tangents parallel to the axis; when A attains a certain
477l'
value which is less than y , the curve acquires a double point ; and as A further
increases, the curve breaks up into two separate ovals, and there are then only two
tangents parallel to the axis, viz. one for the exterior oval and one for the interior
oval. As A continues to increase, the interior oval decreases ; and when A attains
a certain value which is less than nfr* ^^^ interior oval reduces itself to a conjugate
point, and it afterwards disappears altogether. The outside curve is an oval as before,
but smaller and more compressed.
Next, if the directrix touch the dirigent circle, i.e. if a = m\ Then the inside
curve is at first composed of an exterior oval which touches the dirigent circle, and
*)f an interior oval which lies wholly within the dirigent circle. As A increases the
interior oval decreases, reduces itself to a conjugate point, and then disappears. The
outside curve is an oval which always touches the dirigent circle, at first very small
(it may be considered as commencing from a conjugate point corresponding to ^ = 0),
but increasing as A increases negatively.
376 A MEMOIR UPON CAUSTICS. [l45
Next, when the directrix does not meet the dirigent circle, i.e. if a < m*. The
inside curve consists at first of two ovals, an exterior oval lying without the dirigent
circle, and an interior oval lying within the dirigent circle. As A increases the
interior oval decreases, reduces itself to a conjugate point and disappeara The outside
curve is at first imaginary, but when A attains a sufficiently large negative value, it
makes its appearance as a conjugate point, and afterwards becomes an oval which
gradually increases.
Next, when the dirigent circle reduces itself to a point, i.e. if a = 0. The inside
curve makes its appearance as a conjugate point (corresponding to ^ = 0), and as A
increases it becomes an oval and continually increases. The outside curve comports
itself as in the last preceding case.
Finally, when the dirigent circle becomes imaginary, or has for its radius a pure
imaginary distance, i.e. if a is negative. The inside curve is at first imaginary, but
when A attains a certain value it makes its appearance as a conjugate point, and
as A increases becomes an oval and continually increases. The outside curve, as in
the preceding two cases, comports itself in a similar manner.
The discussion, in the present section, of the different forms of the curve is not
a very full one, and a large number of figures would be necessary in order to show
completely the transition from one form to another. The forms delineated in the four
figures were selected as forms corresponding to imaginary values of the parameters bj
means of which the equation of the curve is usually represented, e.g. the equations in
Section xxvin.
XXXVII.
It has been shown that for rays proceeding from a point and refracted at a
circle, the secondary caustic is the Cartesian; the caustic itself is therefore the evohite
of the Cartesian ; this affords a means of finding the tangential equation of the
caustic. In fact, the equation of the Cartesian is
{x'iy' ay + 16^ (a?  m) = ;
and if we take for the equation of the normal
Zf+F7; + ^ = 0,
(where f, t) are current coordinates), then
: a? (ar* f y^  a) H 4.4
: 4ily,
145] A MEMOIR UPON CAUSTICS. 377
equations which give
Z*7» (a;» + y»  a) = ^AZ'XY^
whence eliminating, we have
{Z» + X (mZ'  AX*)Y + 7» (mZ^  AXy  Z»P (oZ + 4ilZ) = 0,
where if, as before, c denotes the radius of the refracting circle, a the distance of the
radiant point from the centre, and fi the index of refraction, we have
A =
d'a
The above equation is the condition in order that the line Xx + Fy H Z = may be
a normal to the secondary caustic (a;" + y' — a)'+16^ (a? — m) = 0, or it is the tangential
equation of the caustic, which is therefore a curve of the class 6 only. The equation
may be written in the more convenient form
xxxvin.
To compare the last result with that previously obtained for the caustic by
reflexion, I write ft = — 1, and putting also c = l and Z = a (for the equation of the
reflected ray was assumed to be Xx + Fy + a = 0), we have
a = a« + 2, A^\a, m = 2^(l + 2a«),
a,nd the equation becomes, after a slight reduction,
*
4a< + 4o'Z (2o» + 1  Z») + (Z» + F») (2o» + 1  X")"  4ia?Y* (a« + 2 + IX) = 0,
^which may be writteu
(2o' + Z (2o« + 1  Z»))« + F» (  40' + 1  Sa'X  2 (2a'' + 1) X' + X*) = ;
this divides out by the fector (X + iy, and the equation then becomes,
(Z»  Z  2a')> + F» ((Z  ly  4a') = 0,
■which agrees with the result before obtained.
378 A MEMOIR UPON CAUSTICS. [145
XXXIX.
Again, to compare the general equation with that previously obtained for parallel
rays refracted at a circle, we must write /a = t> c=1, a = x, Z=^k (for the equation
of the refracted ray was taken to be Xx f Yy + ^ = 0) ; we have then
a^lvk^Vk'a^ A^^k'a^ m = ^ (l +(1 + i»)a*) ,
and, after the substitution, a = x . The equation becomes in the first instance
Jfc« + 2*»Z ~ (!+(!+ k") a«) t«  i**az4 + {X' + F«) — (l + (1 + i») a«) k'  J**(f.Y«J'
and then putting a = x , or, what is the same thing, attending only to the tenuN
which involve a', and throwing out the constant factor k^, we obtain
(X^ + F«)(.Y«  1  k'Y  4^'«F* = 0,
or
x^(X^ii(f^y+Y^{X}i+k)(Xi''k)(x + ik){Xik)~ 0,
which agrees with the former result.
XL.
It was remarked that the ordinary construction for the secondary caustic* could
not be applied to the case of parallel rays (the entire curve would in fact pass off
to an infinite distance), and that the simplest course was to measure the distance
GQ from a line through the centre of the refracting circle perpendicular to tin
direction of the rays. To find the ecjuation of the resulting curve, take the centre of
the circle as the origin and the direction of the incident rays for the axis of jc\ let
the radius of the circle be taken equal to unity, and let fi denote, as before, the
index of refraction. Then if a, /3 arc the coordinates of the point of incidence of a
ray, we have a'^ + y^=l. and considering a, )8 as variable parameters connected bv this
equation, the retjuired curve is the envelope of the circle,
Write now a = cos^, y8 = sin^, then multiplying the equation by —2, and writing
1 + cos 20 instead of 2 cos^ 9, the equation becomes
1 + cos 20  2fi^ (x' b if  2x cos  2i/ sin 5 + 1) = 0,
which is of the form
^cos2^ + 5sin2(9 + C'cose + i)sin^ + A' = 0,
145]
A MEMOIB UPON CAUSTICS.
379
and the values of the coefficients are
4 = 1,
B = 0,
c = v^.
D = Vy.
E = 2fi*(ai' + y*)  2yit»+ 1.
Substituting these values in the equation
12 (il» + £»)  3 (C» + i)*) + 4£>1'
 {27il (C*  D") + BiBCD  (72 (4« + £«) + 9 (C* + i)»)) JE^ + 8£>}' = 0,
the equation of the envelope is found to be
16 {(1  /*» + /*«)  0*' + ,*«) {a? + r/») + At* (a^ + y')*;'
( 4  6/*'  6/** + 4t«
 (6/** + 3/** + Qfif) (a?+y>) 27/*« (a?  yO
(6/*« + 6At')(«' + 2/')'
^ >
S=o,
which is readily seen to be only of the 8tb order. But to simplify the result, write
iirst (a? + y*l) + l, and 2*»l («' + y*l) in the place of a? + y* and a?f respec
tively, the equation becomes
4{(1 /*')» ft* (1 /*») (a;* + y»  1) + /** (a;* + y*  1)"}*
' 2(1 /*•)»
 3/*" (1  A*')* («^ + y*  1)  27/**a?
— <
\
3M*(i/*»)(a!?+yi)»
+ 2/*« (a^ + y»  1)»
>• =0.
/
Write for a moment 1— /a' = 5, ft*(.'c* + y'— l) = p, the equation becomes
or developing,
+ 54 (2g»  39«p  Sgp* + 2/o») ;A*a;«  72 Var* = 0,
and reducing and dividing out by 27, this gives
9«P»(P5)» + 2(p + })(2p})(p23);aV27;aV = 0,
48—2
380 A MEMOIR UPON CAUSTICS. [145
whence replacing q, p by their values, the required equation is
+ 2 (/x« (a;* 4 ya)  2;a« + 1) (2/^2 (a;» + y*)  /*«  1) (/*« (a;« + y«)  2 + ;a«) a;^
which is the equation of an orthogonal trajectory of the refracted rays.
In the case of reflexion, ;a = — I, and the equation becomes
4(a;« + y*l)»27a^=0.
Comparing this with the equation of the caustic, it is easy to see,
Theorem. In the case of parallel rays and a reflecting circle, there is a secondaT"^
caustic which is a curve similar to and double the magnitude of the caustic, t\
position of the two curves differing by a right angle.
XLI.
The entire system of the orthogonal trajectories of the refracted rays might
like manner be determined by finding the envelope of the circle (where, as befo:
a, )8 are variable parameters connected by the equation a"f )9* = 1),
{The result, as far as I have worked it out, is as follows, viz. —
(3  12 [m^ + 'Zmfi^x + ;a* (a:* + y')] + [1  2;a« + 2m*  2/x« {a^ + y»)]*)»
 ([1  2/A« + 2m»  2/x» (a^ + f)] [9 + 18ah« + 36m/A«a: + 18;a* {a^ h f)]
 54 [m» + imfi^x ^ti^ia^ /)]  [1  2;a« + 2iii»  2;a« {a^ + y»)]»)* = 0,
which, it is easy to see, is an equation of the order 8 only. Added Sept. 12. — A. C.j
146.
A MEMOIK ON CUKVES OF THE THIRD ORDER.
[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the
year 1857, pp. 415 — 446. Received October 30, — Read December 11, 1856.]
A CURVE of the third order, or cubic curve, is the locus represented by an
equation such as U={*^x, y, ^)» = 0; and it appears by my "Third Memoir on
Quantics," [144], that it is proper to consider, in connexion with the curve of the third
order 17=0, and its Hessian J? 17=0 (which is also a curve of the third order), two
curves of the third class, viz. the curves represented by the equations PU=0 and QU=0.
These equations, I say, represent curves of the third class; in fact, PU and QU are
contravariants of J7, and therefore, when the variables x, y, z o{ U are considered as
point coordinates, the variables f, 17, (f of PJ7 and QU must be considered as line
coordinates, and the curves will be curves of the third class. I propose (in analogy
with the form of the word Hessian) to call the two curves in question the Pippian
and Quippian respectively. [The curve PU=0 is now usually called the Cayleyan.]
A geometrical definition of the Pippian was readily found; the curve is in fact Steiner's
curve i2o mentioned in the memoir "AUgemeine Eigenschafben der algebraischen Curven,"
Crelle, t. XLVii. [1854] pp. 1 — 6, in the particular case of a basiscurve of the third
order; and I also found that the Pippian might be considered as occurring implicitly
in my "M^moire sur les courbes du troisifeme ordre," Liouville, t. IX. [1844] pp.
285 — 293 [26] and "Nouvelles remarques sur les courbes du troisi^me ordre," Liouville,
t. X. [1845] pp. 102 — 109 [27]. As regards the Quippian, I have not succeeded in
obtaining a satisfactory geometrical definition ; but the search aft^r it led to a variety
of theorems, relating chiefly to the firstmentioned curve, and the results of the investi
gation are contained in the present memoir. Some of these results are due to Mr
Salmon, with whom I was in correspondence on the subject. The character of the
results makes it difficult to develope them in a systematic order; but the results
are given in such connexion one with another as I have been able to present them
382 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46
in. Considering the object of the memoir to be the establishment of a distinct
geometrical theory of the Pippian, the leading results will be found summed up in
the nine different definitions or modes of generation of the Pippian, given in the con
cluding number. In the course of the memoir I give some further developments
relating to the theory in the memoirs in Liouville above referred to, showing its
relation to the Pippian, and the analogy with theorems of Hesse in relation to the
Hessian.
Article No. 1. — Definitions, <tc.
1. It may be convenient to premise as follows: — Considering, in connexion with
a curve of the third order or cubic, a point, we have :
(a) The first or conic polar of the point.
(b) The second or line polar of the point.
The meaning of these terms is well known, and they require no explanation.
Next, considering, in connexion with the cubic, a line —
(c) The first or conic polars of each point of the line meet in four points,
which are the four poles of the line.
{d) The second or line polars of each point of the line envelope a conic, which
is the lineO'polar envelope of the line.
And reciprocally considering, in connexion with a curve of the third class, a line^
we have:
{e) The first or conic pole of th^ line.
(/) The second or pointpole of the line.
And considering, in connexion with the curve of the third class, a point —
(g) The first or conic poles of each line through the point touch four linefi,
which are the four polars of the point.
(h) The second or point poles of each line through the point generate a conic
which is the pointpole lociis of the point.
But I shall not have occasion in the present memoir to speak of these reciprocal
figures, except indeed the first or conic pole of the line.
The term conjugate poles of a cubic is used to denote two points, such that the
first or conic polar of either of them, with respect to the cubic, is a pair of lines
passing through the other of them. Reciprocally, the term conjugate polars of a curve
of the third class denotes two lines, such that the first or conic piole of either of
them, with respect to the curve of the third class, is a pair of points lying in the
other of them.
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 383
The expression, a ayzygetic cubic, used in reference to two cubics, denotes a curve
of the third order passing through the points of intersection of the two cubics; but
in the present memoir the expression is in general used in reference to a single cubic,
to denote a curve of the third order passing through the points of intersection of
the cubic and its Hessian. As regards curves of the third class, I use in the memoir
the full expression, a curve of the third class syzygetically connected with two given
curves of the third class.
It is a wellknown theorem, that if at the points of intersection of a given line
with a given cubic tangents are drnwn to the cubic, these tangents again meet the
cubic in three points which lie in a line; such line is in the present memoir
termed the satellite line of the given line, and the point of intersection of the two
lines is termed the satellite point of the given line; the given line in reference to
its satellite line or point is termed the primary line.
In particular, if the primary line be a tangent of the cubic, the satellite line
coincides with the primary line, and the satellite point is the point of simple inter
section of the primary line and the cubic.
Article No. 2. — Oroup of Theorems relating to the Conjugal Poles of a Vubic,
2. The theorems which I have first to mention relate to or originate out of the
theory of the conjugate poles of a cubic, and may be conveniently connected together
and explained by means of the accompanying figure.
The point ^ is a point of the Hessian ; this being so, its first or c(mic polar,
with respect to the cubic, will be a pair of lines passing through a point F of the
Hessian ; and not only so, but the first or conic polar of the point F, with respect
to the cubic will be a pair of lines passing through E. The pair of lines through
384 A MEMOIR ON CURVES OF THE THIRD ORDER. [146
F are represented in the figure by FBA, FDC, and the pair of lines through E are
represented by EC A, EDC, and the lines of the one pair meet the lines of the other
pair in the points A, B, C^ D. The point 0, which is the intersection of the lines
AD, BC, is a point of the Hessian, and joining EOy FO, these lines are tangents to
the Hessian at the points E, F, that is, the points E, F are corresponding points of
the Hessian, in the sense that the tangents to the Hessian at these points meet in
a point of the Hessian. The two points E, F are, according to a preceding definition,
conjugate poles of the cubic.
The line EF meets the Hessian in a third point 0, and the points G, are
conjugate poles of the cubic. The first or conic polar of 0, with respect to the cubic,
is the pair of lines AOD, BOC meeting in 0. The first or conic polar of 0, with
respect to the cubic, is the pair of lines QEF and Gfeff! meeting in C The four
poles of the line EO, with respect to the cubic, are the points of intersection of the
first or conic polars of the two points E and 0, that is, the four poles in question
are the points jP, F, e, e'. Similarly, the four poles of the line FO, with respect to
the cubic, are the points E, E, /, f.
The line EF, that is, any line joining two conjugate poles of the cubic, is a tangent
to the Pippian, and the point of contact V is the harmonic with respect to the points
E, F (which are points on the Hessian) of G, the third point of intersection with
the Hessian. Conversely, any tangent of the Pippian meets the Hessian in three
points, two of which are conjugate poles of the cubic, and the point of contact is the
harmonic, with respect to these two points, of the third point of intersection with
the Hessian.
The line GO in the figure is of course also a tangent of the Pippian, and more
over the lines FBA, FDC (that is, the pair of lines which are the first or conic polar
of E) and the lines EGA, EDB (that is, the pair of lines which are the first or
conic polar of F) are also tangents to the Pippian. The point E represents any
point of the Hessian, and the three tangents through E to the Pippian are the line EFG
and the lines EGA, EDB; the line EFG is the line joining E with the conjugate
pole F, and the lines EGA, EDB are the first or conic polar of this conjugate pole
F with respect to the cubic. The figure shows that the line EG (the tangent to
the Hessian at the point E) and the beforementioned three lines (the tangents
through E to the Pippian), are harmonically related, viz. the line EG the tangent of
the Hessian, and the line EF one of the tangents to the Pippian, are harmonics
with respect to the other two tangents to the Pippian. It is obvious that the
tangents to the Pippian through the point F are in like manner the line GFE, and
the pair of lines FBA, FBG, and that these lines are harmonically related to FO the
tangent at F of the Hessian. And similarly, the tangents to the Pippian through
the point are the line GO and the lines AOD, BOO, and the tangents to the
Pippian through the point G are the line GO and the lines GFE and Gfefe', Thus
all the lines of the figure are tangents to the Pippian except the lines EG, FO,
which are tangents to the Hessian. It may be added, that the lineopolar envelope
of the line EF with respect to the cubic is the pair of lines OE, OF,
146 J A MEMOIR ON CURVES OF THE THIRD ORDER. 385
It will be presently seen that the analytical theory leads to the consideration of
a line IJ (not represented in the figure): the line in question is the polar of E
(or F) with respect to the conic which is the first or conic polar of F (or E) with
respect to any syzygetic cubic. The line /J" is a tangent of the Pippian, and more
over the lines EF and // are conjugate polars of a curve of the , third class
syzygetically connected with the Pippian and Quippian, and which is moreover such
that its Hessian is the Pippian.
Article Nos. 3 to 19. — Analytical investigations, comprising the proof of the
theoremSy Article No. 2.
3. The analytical theory possesses considerable interest. Take as the equation of
the cubic,
Z7 = ir' + y* + '^*+ ^locyz = ;
then the equation of the Hessian is
HU= f(af + f + z')(\ + W) xyz^O;
and the equation of the Pippian in line coordinates (that is, the equation which
expresses that ^x+r)y^ ^z = is a tangent of the curve) is
Pir=  Z(» h 7;»+ ?») + ( 1 + 4P) ?i7?= 0.
The equation of the Quippian in line coordinates is
Qir= (1  lOP) (f h 7;» + ?»)  61^ (5 + 4P) fi;C= ;
and the values of the two invariants of the cubic form are
T=120P8Z«,
values which give identically,
2^64fll* = (l+8Z'')»;
the lastmentioned function being in fact the discriminant.
4. Suppose now that (X, Y, Z) are the coordinates of the point E, and
(X\ Y, Z') the coordinates of the point F; then the equations which express that
these points are conjugate poles of the cubic, are
XT{l(YZ'^rZ) =0,
7Fhi(ZX'+^Z) = 0,
ZZ' +1{XY + X'Y)=0;
and by eliminating from these equations, first (X\ Y, Z\ and then (X, F, Z), we find
ZH^* + F» + Z»)(l+2P)XFir =0,
p (X» + F» + Z'O  (1 + 2P) X'YZ' = 0,
which shows that the points Ey F are each of them points of the Hessian.
C. II.
49
386 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46
5. I may notice, in psussing, that the preceding equations give rise to a somewhat
singular imaymmetrical quadratic transformation of a cubic form. In hct, the second
and third equations give X' : T : Z'^YZPX^ : PX7l^ : PZXlYK And sub
stituting these values for X\ Y, Z' in the form
Z«(Z'»+ F» + Z'»)(l + 2l')X'rZ',
the result must contain as a factor
f(Z»+ F» + Z»)  (1 + 2P) XTZ\
the other &ctor is easily found to be
P(P(Z«+F» + 2*) + 3ZZFZ).
Several of the formulsB given in the sequel conduct in like manner to unsymmetrical
transformations of a cubic form.
6. I remark also, that the lastmentioned system of equations gives, symmetricaUy,
X'* : y : Z" : FZ' : Z'X' : X'T
= YZl*X* : ZXl'Y^ : XTPZ* : 1*YZIX* : 1*ZXIY* : PXYW;
and it is, I think, worth showing how, by means of these relations, we pass from
the equation between X', Y', Z! to that between X, Y, Z. In fact, representing, for
shortness, the foregoing relations by
X'* : T* : Z* : TZ' : Z'X' : X'T ^ A . B : G : F : Q : H,
we may write
X' = AF=GH, T = BQ = HF, Z' = CH = FQ, ABC = FGH;
and thence
X'*='AF.G'H\ T* = BO.H*F^, Z'* = CH.F>G', X'TZ^F'O'H*;
hence
l^iX'* + T* + Z*)(l+2l*)X'TZ' = FGH{l^iAQH+BHF+CFQ)(l + 2i')FOH}.
But we have
l^(AOH + BHF+ CFG) =  (21' + 1») (X* + F* + Z») XYZ+{1* + 2^)(Y*Z* + Z'X* + X*Y*),
(l + 2lf)FGH = (^ + 2l!>)(X*+Y* + Z*)XYZ+(l*+2P)(Y*Z* + Z*X* + X*Y')
+ l'(ll^)(l + 2l*)X'Y*Z';
and thence
P (AGH+BHF+ CFG)  (1 + 21') FGH
= _ p(l _ P) {P(Z«+ Y*+Z*)XYZ (1 + 2l^)X'Y*Z'} ;
and finally,
l'(X"+ Y'* + Z'')(l + 2l')X'Y'Z' = l'{l + l*)(lYZX'){lZX Y*)(IXYZ')XYZ
>i{l*(X'+Y*\Z*){l+2fi)XYZ\.
146] A MEMOnt ON CURVES OF THE THIRD ORDER. 387
We have also, identically,
ABGFQH = j(l+l*)XYZ{l*(X*\Y* + Z>)(l + 2l*)XYZ},
which agrees with the relation ABG — FGH=0.
7. Before going further, it will be convenient to investigate certain relations
which exist between the quantities (X, Y, Z), (X', Y, Z), connected as before by
the equations
XX' + liYZ' +TZ) =0,
YT + 1 {ZX' + Z'X) = 0,
ZZ' +i(XF' + Z'F) = 0,
and the quantities
^=YZ'TZ, a=XX' = j(YZ + TZ),
v=zx'z'x, fi=Yr=jizx'+z'X),
r = Zr  Z'F, y=ZZ' = (XT + X'Y).
We have identically,
2XX'(YZ'TZ) + (XT + X'Y)(ZX'ZX) + {ZX' + Z'X)(XTX'Y) = 0;
or expressing in terms of f, 17, ^, a, /9, 7 the quantities which enter into this
equation, and forming the analogous equations, we have
2K fV /9r=0, (A)
y^+2lfir, ar=0,
/S o,, + 2tyi:=0.
We have also
X*rZX''YZ = i{(XT + X'Y)(ZX'Z'X) + {ZX' + Z'X)(XY'X'Y)},
and thence in like manner,
X»rZ'  X"YZ = ^ (7^  /30, (B)
Z*X'r  X'*YZ = i (j8f  ar,l
Again, we have
( YZ'  rzy ={YZ'+ YZf  4 ytzz',
(ZX' Z'X)(Xr  X'Y)^(ZX' + ZX)(Xr + X'Y) + 2XX' (YZ' + TZ);
49—2
388 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46
and thence
p= i«'4/87, (C)
1;*= ,/3»4ya,
and conversely
J(l + 8P)«'= ?4?.;r. (D)
l(l + 8P)7a=2V+ ?f.
J,(l + 8P)a/9 = 2ZC»+ f,.
8. It is obvious that
ix + fiy + J^ =
is the equation of the line EF joining the two conjugate poles, and it may
shown that
aa? + /3y + 7? =
is the equation of the line /J", which is the polar of E with respect to a con ^
which is the first or conic polar of F with respect to any syzygetic cubic. In fitc^
the equation of a syzygetic cubic will be aj' + y*+^H6Xa;y? = 0, where X is arbitrar]!^
and the equation of the line in question is
or developing,
XTx+YYy + ZZ'z
•\\{YZ''^YZ)x\{ZT^Z'X)y + {XT + TY)z]^0\
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 389
and the function on the lefthand side is
which proves the theorem.
9. The equations (A) by the elimination of (f, % f), give
Z(a» + /S»+7») + (H4Z») 0^7=0,
which shows that the line IJ is a tangent of the Kppian : the proof of the theorem
is given in this place because the relation just obtained between a, /9, 7 is required
for the proof of some of the other theorems.
10. To find the coordinates of the point in which the line EF joining two
conjugate poles again meets the Hessian.
We may take for the coordinates of G,
and, substituting in the equation of the Hessian, the terms containing u^ 1^ disappear,
and the ratio u \ v \b determined by a simple equation. It thus appears that we
may write
V = 3P(Z»Z' f Y^T + Z^Z'){\ f 2P) {YZT + ZXY h XYZ') ;
hence introducing, as before, the quantities f, 17, f, a, /9, 7, we find
uZ + vX = 3P (717  )80 + (1 + 2P) (ST'FZ'  Z'«F^ ;
but from the first of the equations (B),
and therefore the preceding value of uX^vX' becomes
which is equal to
—21— (y't  ^^^
Hence throwing out the constant factor, we find, for the coordinates of the point (?,
the values
71; ^f, af7f, ^fai7.
11. To find the coordinates of the point 0.
Consider as the point of intersection of the tangents to the Hessian at the
points E, Fy then the coordinates of are proportional to the terms of
3PZ»  lf 2PFZ , ZM^  lf n^ZX , 3?2?  1 + 2PZF
3PZ'«H2Z»F^', 3PF«1h2Z»Z'Z', 3PZ'« 1 f 2PZ'F'
390 A MEMOIR ON CURVES OF THE THIRD ORDER. [146
Hence the a;coordinate is proportional to
which is equal to
9i*(F»Z'«F«2?)l3P(l + 2P)FF(XFZ'r) + 3P(l + 2P)ZZ'(^Z'Z'Z)
(1 + 2P)» ZZ'(FZ'rZ);
or introducing, as before, the quantities f, 17, f, a, /3, 7, to
 9Paf + 3P (1 + 2P) (/9?+ w)  (1 + Wy of,
= ( 1  13/»  «•) af + 2Z« (1 + 2Z») (/9? + 71;).
But by the first of the equations (A) /3f + 71; = 2Zaf , and the preceding value thus
becomes ( — 1 — 7P + 8P) af. Hence throwing out the constant factor the coordinates of
the point are found to be
af fiVy 7?
12. The points (?, are conjugate poles of the cubic.
Take a, 6, c for the coordinates of G, and a', h\ c* for the coordinates of 0, we have
a, 6, c =r7^?, a?7f /3fa^,
a', 6', C = af , /9i7 , 7?.
These values give aa' + i (6c' + 6'c)
= «f(r7/8?) + M/8'7(/8f«'7) + 7?K7f)}
= ^7(«7 + W + '7"(~^«^) + r*(M + f?(«/8V);
or substituting for f?/, 17', f ^ f f their values in terms of a, /9, 7, this is
+ ( f 47a)( /a/8)
+ ( ^ 4ay9)( /a7)
+ (f/9«ia7)(~«/8V),
which is identically equal to zero. Hence, completing the sjrstem, we find
aa' + i (6c' + 6'c) = 0,
W h I {ca' + c'a) = 0,
co' + i (a6' + a'6) = 0,
equations which show that (as well as G) is a point of the Hessian, and that the
points Qy are corresponding poles of the cubic.
146] A MEMOm ON CUBVES OF THE THIRD OBDEB. 391
13. The line EF joining a pair of conjugate poles of the cubic is a tangent of
the Pippian^
In fact, the equations (A), by the elimination of a, /8, 7, give
f(P + ^*+?»)l(l+4P)^7?=0,
which proves the theorem.
14. To find the equation of the pair of lines through F, and to show that these
lines are tangents of the Pippian.
The equation of the pair of lines considered as the first or conic polar of the
conjugate pole E, is
Z (a;» + nyz) + Y{y^ + 2lzx) + Z(z* + 2lxy) = 0.
Let one of the lines be
Xa? f A^y + v^ = 0,
then the other is
X Y Z ^
and we find
2lXfiv  Fi/»  Z/i« = 0,
 Xv" + 2lYv\  ^« = 0,
 Z/A«  yx« + 2iZfiv = 0,
any two of which determine the ratios X, /a, 1/.
The elimination of Z, F, Z gives
2Z/A1;, i/»,  /A« =0,
 v" , 2,lv\  X«
 /A« ,  XS 2iX/i
which is equivalent to
X/ii; { Z(X» + /i« + !;») + ( 1 + 4P) X/ii/} = 0;
or, omitting a factor, to
 /(X» + /i« + !/») + ( 1 +4i»)X/ii; = 0,
which shows that the line in question is a tangent of the Pippian.
15. To find the equation of the pair of lines through 0,
The equation of the pair of lines through ^ is in like manner
Z'(a;* + 2iy2r)+ F(y« + 2i«a?) + Z'(^« + 2Ziry) = 0;
^ Steiner's onrre Rf^ in the partioalar oase of a onbio baaisonrve, is aooording to definition tiie envelope
of the line EF^ that is, the onrve R^ in the partionlar case in question is the Pippian.
392 A MEMOIIt ON CURVES OF THE THIRD ORDER. [l46
and combining this with the foregoing equation,
X{a? + 2lyz) f F(y' f 2lzx) + Z(z^ + 2lxy) =
of the pair of lines through F, viz. multiplying the two equations by
X^X' h PF 4 Z^Z\  {XX'^ + YT^ + ZZ'%
and adding, then if as before
we find as the equation of a conic passing through the points A, B, C, D, the equation
a (ic» + 2lyz) + h{y^ + 2lzx) + c{z'^ + 2lxy) = 0.
But putting, as before,
then a', 6', c' are the coordinates of the point 0, and the equations
aa' + i(6c' + 6'c) = 0,
66' + /(ca'+c'a) = 0,
show that the conic in question is in fact the pair of lines through the point 0.
16. To find the coordinates of the point F, which is the harmonic of with
respect to the points E, F.
The coordinates of the point in question are
uXvT, uYvT, uZvZ',
where u, v have the values given in No. 10, viz.
u = 3i»(ZZ'«+ 77'« + ZZ'0 + (l + 2P)(FZ'X + ^Z'F + X'rZ),
V = 3Z« (Z»X' + PF' + Z^Z')  (1 + 2P) (YZX' + ZXY' + XYZ") ;
these values give
uX  vZ' =  81' {2Z»Z'« + (Zr + Z' F) YY' + (ZZ' + Z'Z) ZZ]
+ (l + 2l^){(XY' + X'Y)(XZ'\X'Z) + Xr(YZ'+Y'Z]]
and therefore
iiZt;Z' = 3p2a«?/37 ■f(l + 2i«)iy97Ja«.
= J,(l+8?)(Za> + /87);
and consequently, omitting the constant factor, the coordinates of F may be taken to be
fa« + /97, //3» + 7a, V + o^
146] A MEMOIR ON CURVES OP THE THIRD ORDER. 393
17. The line through two consecutive positions of the point F is the line EF.
The coordinates of the point T are
•la^^l3y, ^l^ + ya, ^Irf + a/S;
and it has been shown that the quantities a, /9, 7 satisfy the equation
 Z (a» f i8» 4 7») + ( 1 + 4P) a/97 = 0.
Hence, considering a, /9, 7 as variable parameters connected by this equation, the
equation of the line through two consecutive positions of the point F is
I ^si(^ + {l + 4l^)fiy, 3//S«l(l+4P)7a,  SZ^h ( 1 h 4P) a/3 > = ;
^, 2la , 7 , /3
y, 7 , 2Z/9 . a
z, /3 , OL , 2/7
and representing this equation by
Lx + My {• Nz = 0,
we find
Z= (4P/37  a») ( 3/a» + ( lf 4/0/87)
+ (a/9+ 2/7') ( n^ + ( 1 + 4P) 7a)
+ (07 + 2/^) ( 3^y^ +(lh4P)a/9);
or, multiplying out and collecting,
L = 3Za* h ( 1  8/») a?Py h ( 5/ + 8P) (a/8» h a7») f ( 16/» + 16/») /3V ;
but the equation
 Z (a» f y3» f y) + (  1 + 4/^ 0/87 =
gives
3^0* =  3i (ay8« + OT*) + ( 3 + 12i») tf»/97,
and we have
Z = ( 4 + 4P) OL^Py + ( 8i + SV) (o/S^h ay) + ( 16P + 16Z0/3V
= ( 4 + 4Z») (a»/37 + 2Z (a5» h a^) + 4Z»/8V)
= (_ 4 + 4i.) (a^ + 2/^) (a/9 h 2ty») ;
or, in virtue of the equations (D),
Z = (4 + 4P)Z«5f .Pfi7 = (4 + 4P)i*fi7?=(4h4/»)/*fi;?.f.
Hence, omitting the common factor, we find L : M : N=^ : r) : ^, and the equation
Lx + My + Nz = becomes
f a? + w + C^ = 0,
C. II.
50
394 A MEMOIB ON CUBYES OF THE THIRD ORDER. [l46
which is the equation of the line EF, that is, the line through two consecutive positioDs
of r is the line EF\ or what is the same thing, the line EF touches the Pippian
in the point F which is the harmonic of Q with respect to the points E, F.
18. The lineopolar envelope of the line EF, with respect to the cubic, is the
pair of lines OE, OF.
The equation of the pair of lines OE, OF, considered as the tangents to the
Hessian at the points E, F, is
r=0.
X {(3PZ'«  1 + 2PFZ0 X f {ZP r*  1 + WZ'X') y + (3P^'*  1 + 2PX'Y') z] )
Here on the lefthand side the coefficient of a;* is
9/*Z«Z'«  3? (lf 2V) (X'Y'Z' + X'^YZ) + (1 + 2P)« YTZ^,
which is equal to
that is
91^  3P (1 + 2Z») (Z«/97 + ^ a«) + (1 h 2Z»)» /Sy,
j(/ + ?){3ia»+2(lh2P)/37};
and the coefficient of yz is
9^(F«Z'»+ F«2?) 3Z«(1 + 2V) {YY' {XY' + X'Y)^ ZZ'iXZ' ^^X'Z))
h(l+2P)«ZZ'(F^+F'Z),
which is equal to
that is
J(/l/*){(l4/»)a«6P/97)
Hence completing the system and throwing out the constant factor, the equation of
the pair of lines is
(3k«2(H2Z»)/87, 3/i8« + 2 (1 + 2P) 7a, 3^ + 2 (1 h 2Z') a/9.
(1 » 4P) a«  6/^/87, (1  4P) y3«  QPyOL, (1  4P) 7*  &Pol^\x, y, ^)« = 0.
But the equation of the line EF is a? h ^y + f^ = 0, and the equation of its liueopolar
envelope is
f, X, Iz. ly
f), Iz, y, Ix
?, ly, X , z
= 0;
146] A MEMOIR ON CUBVES OF THE THIBD OBDEB. 395
or expanding,
or arranging in powers of x, y, z,
(/«p2/i7r, tvm. ^"^2i?i7, ip+^;?, hv'+m> iP+mii^,y>^y^o:
and if in this equation we replace f , &c. by their values in terms of a, fi, y, aa
given by the equations (D), we obtain the equation given as that of the pair of lines
OE, OF.
19. It remains to prove the theorem with respect to the connexion of the lines
EF, TJ.
The equations (A) show that the two lines
^x+Tfy +5e^ = 0,
ax\fiy+^=0,
(where f, i;, f and a, /3, y have the values before attributed to them) are conjugate
polars with respect to the curve of the third class,
in which equation ; 17, ^ denote current line coordinates. The curve in question is of
the form APV +BQV = 0. We have, in fact, identically,
It is clear that the curve in question must have the curve PCr = for its Hessian;
and in fact, in the formula of my Third Memoir, [144]
H{6aPU + fiQU)=^(2T, 48S*, ISTS , T*+16S»$a, fiyPU
+ ( 8>8f, r,8S*, ^T8 Ha, fiYQU,
the coefficient of QCT is
and therefore, putting a = JT, ^ = — 4/S, we find
JT(3r.Pcr4S.<2J7) = H3^645*)»Pcr.
Article No. 20. — Theorem relating to the curve of the third doss, mentioned in the
preceding Article,
20. The consideration of the curve ST.PU— 4aS.QU=0, gives rise to another
geometrical theorem. Suppose that the line (^, 17, (f), that is, the line whose equation
is ^ + rfy + ^z = 0, is with respect to this curve of the third class one of the four
polars of a point {X, T, Z) of the Hessian, and that it is required to find the envelope
of the line ^x + r)y+^z = 0,
50—2
396 A UEHOIB ON CURVES OP THE THIRD ORDER. [U6
We have
X: y,^ip,j;: Vff^ifft,
and X, y, Z are to be eliminated from these equations, and the equation
PiX*+V' + Z*){l+2l')XrZ=0
of the Hessian. We have
x'+Y'+z'= Hp+v'+^y
+ 91 fVf
(l+2P)(i,'f + rP + ?;*).
xYz= i(p+y+c.)f,{:
4 (1+1')^^
^(VS' + rP + P'j').
and thence
nu= P (^+^> + ^y
(i + 5l^)(P + v' + n^?
+ (I + 101' 21*)^^;
and equating the righthand side to zero, we have the equation in line coordinates of
the curve in question, which is therefore a curve of the sixth class in quadratic
syzygy with the Pippian and Quippian.
Articl« No. 21. — Geometrical definition of the Quippian,
21. I have not succeeded in obtaining any good geometrical definition of the
Quippian, and the following is only given for want of something better.
T. PU \PeH(aU + 6^HU)\  P{6HU) [T{aU+&^HU) .P{aU+ Q^HU)] =0,
which is derived in what may be taken to be a known manner from the cubic, is ii
general a curve of the sixth class. But if the syzygetic cubic aU + 60HV = be
properly selected, viz. if this curve be such that its Hessian breaks up into three
lines, then both the Pippian of the cubic aU+6^HU = 0, and the Pippian of iU
Hessian will break up into the same three points, which will be a portion of the
curve of the sixth class, and discarding these three points the curve will sink down
to one of the third class, and will in fact be the Quippian of the cubic.
To show this we may take
an + 60HU = iX!'+y' + A=O
146] A MEMOIIt ON CURVES OF THE THIRD ORDER. 397
as the equation of the syzygetic cubic satisfying the prescribed condition, for this value
in fact gives
H (aU^efiHU) =  xyz, = 0,
a system of three lines. We find, moreover,
P(aU + 6fiHU) = P(x' + f+z'l:=''^^
and
P {6H{aUh 6l3HU)]^P{''6ayz), = 4,^^,
the latter equation being obtained by first neglecting all but the highest power of I in
the expression of PU, and then writing l = —l: we have also T(aU\6l3HU)=l.
Substituting the above values, the curve of the sixth class is
^^^^T.PU+P(6HU)]=0',
or throwing out the factor fi/Jf, we have the curve of the third class,
^T.PU+P(6HU) = 0.
Now the general expression in my Third Memoir, viz.
P {aU+ 60HU) = (a» + USafi' + *T^) PU ¥ (c^fi  48/3*) QU,
putting a = 0, /8 = 1, gives
P (6HU) = iT.PUi8.QU,
or what is the same thing,
4T.PU + P{6HU) = 4>8,QU;
and the curve of the third class is therefore the Quippian QU = 0, It may be remarked,
that for a cubic Cr = the Hessian of which breaks up into three lines, the above
investigation shows that we have Pf/^= — fi?^, P(6ffJ7) = — 4fi7f. and T=l, and conse
quently that ^T.PU { P{6HU) ought to vanish identically; this in fact happens in
virtue of the factor 8 on the righthand side, the invariant iS of a cubic of the form
in question being equal to zero ; the appearance of the factor 8 on the righthand
side is thus accounted for d priori.
Article No. 22. — Theorem relating to a line which meets three given conies in six points in
involution.
22. The envelope of a line which meets three given conies, the first or conic
polars of any three points with respect to the cubic, in six points in involution, is
the Pippian.
It is readily seen that if the theorem is true with respect to the three conies,
dx ' dy * dz *
398 A M£MOIB ON CURVES OF THE THIBD ORDER. [l46
it is true with respect to any three conies whatever of the form
^ dU , dU dU ^
that is, with respect to any three conies, each of them the first or conic polar of
some point (X, /a, v) with respect to the cubic. Considering then these three conies,
take ^x+riy\' ^z = as the equation of the line, and let (Z, F, ^ be the coordinates
of a point of intersection with the first conic, we have
fZfi7F+?Z=0,
X^+21YZ =0;
and combining with these a linear equation
aX{l37+yZ=0,
in which (a, /3, 7) are arbitrary quantities, we have
and hence
an equation in (a, /3, 7) which is in fact the equation in line coordinates of the two
points of intersection with the first conic. Developing and forming the analogous
equations, we find
( V' > r , 2Za ^rf , Iv^ , fiyif'ia, A 7)" = 0,
which are respectively the equations in line coordinates of the three pairs of intersectionB.
Now combining these equations with the equation 7 = 0, we have the equations
of the pairs of lines joining the points of intersection with the point (« = 0, y = OX and
if the six points are in involution, the six lines must also be in involution, or the
condition for the involution of the six points is
2^r, ^ , m =0.
that is,
4P:«^; ( f7  ^D + Vr + ^f?* + 2i»Vr + Sf'lVC + C* ( f^  ^D = ;
or, reducing and throwing out the fsictor ^, we find
K?+v + r)+(n*i')fv?=o,
which shows that the line in question is a tangent of the Pippian.
146] A M£MOIB ON CURVES OF THE THIRD ORDER. 399
It is to be remarked that any three conies whatever may be considered as the
first or conic polars of three properly selected points with respect to a properly selected
cubic curve. The theorem applies therefore to any three conies whatever, but in this
case the cubic curve is not given, and the Kppian therefore stands merely for a curve
of the third class, and the theorem is as follows, viz. the envelope of a line which
meets any three conies in six points in involution, is a curve of the third class.
Article No. 23. — Completion of the theory in Liouville, and comparison ivith analogous
theorems of Hesse.
In order to convert the foregoing theorem into its reciprocal, we must replace the
cubic U=0 by a curve of the third class, that is we must consider the coordinates
which enter into the equation as line coordinates ; and it of course follows that the
coordinates which enter into the equation PU = must be considered as point
coordinates, that is we must consider the Pippian as a curve of the third order: we
have thus the theorem; The locus of a point such that the tangents drawn from it
to three given conies (the first or conic poles of any three lines with respect to a
curve of the third class) form a pencil in involution, is the Pippian considered as a
curve of the third order. This in fact completes the fundamental theorem in my
memoirs in Liouville above referred to, and establishes the analogy with Hesse s results
in relation to the Hessian ; to show this I set out the two series of theorems as
follows :
Hesse, in his memoirs On Curves of the Third Order and Curves of the Third
Class, CreUe, tt. xxviii. xxxvi. and xxxviii. [1844, 1848, 1849], has shown as follows :
(a) The locus of a point such that its polars with respect to the three conies
jr = 0, F=0, Z=0 (or more generally its polars with respect to all the conies of the
series \X ^ /mY •{• pZ =^0) meet in a point, is a curve of the third order F = 0.
(fi) Conversely, given a curve of the third order F=0, there exists a series of
conies such that the polars with respect to all the conies of any point whatever of
the curve F=0, meet in a point.
(7) The equation of any one of the conies in question is
dx dy dz *
that is, the conic is the first or conic polar of a point (X, /a, v) with respect to a
certain curve of the third order 17'= 0; and this curve is determined by the condition
that its Hessian is the given curve F=0, that is, we have V=HU.
(S) The equation V^HU is solved by assuming U = aV+bHV, for we have then
H (aV'\bHV)=^AV + BHV, where A, B are given cubic functions of a, 6, and thence
V=sHU=AV+BHV, or 4 = 1,5 = 0; the latter equation gives what is alone important,
the ratio a : b; and it thus appears that there are three distinct series of conica,
400 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46
each of them having the abovementioned relation to the given curve of the third
order F=0.
In the memoirs in Lioumlle above referred to, I have in effect shown that —
(a') The locus of a point such that the tangents from it to three conies, repre
sented in line coordinates by the equations X = 0, F=0, Z = (or more generally with
respect to any three conies of the series \X + fiT\'vZ=0) form a pencil in involution,
is a curve of the third order V=0.
(^) Conversely, given a curve of the third order F = 0, there exists a series of
conies such that the tangents from any point whatever of the curve to any three of
the conies, form a pencil in involution.
Now, considering the coordinates which enter into the equation of the Pippian as
point coordinates, and consequently the Pippian as a curve of the third order, I am
able to add as follows :
(7') The equation in line coordinates of any one of the conies in question is
^ dU dU dU ^
that is, the conic is the first or conic polar of a line (X, fi, v) with respect to a
certain curve of the third class U = ] and this curve is determined by the condition
that its Pippian is the given curve of the third order F = 0, that is, we have
V=PU.
(Sf) The equation F = Pf7' is solved by assuming U=aPV+bQVy for we have
then P(aPV\'bQV)==AV'^BHVy where A and B are given cubic functions of a, i;
and thence V=^PU = AV+ BHV, or ^=1, 5 = 0; the latter equation gives what is
alone important, the ratio a : b; and it thus appears that there are three distinct
curves of the third class [7=0, and therefore (what indeed is shown in the Memoirs
in Liouville) three distinct series of conies having the abovementioned relation to the
given curve of the third order F= 0.
It is hardly necessary to remark that the preceding theorems, although precisely
analogous to those of Hesse, are entirely distinct theorems, that is the two series are
not connected together by any relation of reciprocity.
Article Nos. 24 to 28. — VarioiLS investigations and theorems.
24. Reverting to the theorem (No. 18), that the lineopolar envelope of the line
EF is the pair of lines OE, OF; the line EF is any tangent of the Pippian, hence
the theorem includes the following one:
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 401
The lineopolar envelope with respect to the cubic, of any tangent of the Pippian,
is a pair of lines.
And conversely,
The Pippian is the envelope of a line such that the lineopolar envelope of the
line with respect to the cubic is a pair of lines.
It is I think worth while to give an independent proof. It has been shown that
the equation of the lineopolar envelope with respect to the cubic, of the line
f J h lyy + Jf^ = (where f , i;, Jf are arbitrary quantities), is
(_/.^_2f,f, ihf2i^. p^2i^, ir+^*^r. W+m if'+PM^. y. ^)'=o;
and representing this equation by
i (a, 6, c, /, g, h\x, y, zf = 0,
we find
he /« = f ( r + 8ZV+ S^'?" + 12Pf«7?)>
a6  A« = ? (8Pf + 8ZY  ?* + 12f^^70>
gh  a/= f {21' (^ + ^ + f 3) + 4; (1 + 2P) fi7?) + (1 + 8P) 7;«?«,
A/6i7 = ^(2P(r + ^+?^)+4Kl + 2P)f^?)h(l + 8P)rr,
/flrcA = r(2/»(r + y + f') + «(lh2?)^?)h(l+8Z')r^^
and after all reductions,
ahcaf'hg'ch?^2fgh
or the condition in order that the conic may break up into a pair of lines is PU=0.
25. The following formulae are given in connexion with the foregoing investigation,
but I have not particularly considered their geometrical signification. The lineopolar
envelope of an arbitrary line ^x{r)y\'^z=0, with respect to the cubic
has been represented by
(a, b, c, /, g, A$a?, y, ^)' = ;
and if in like manner we represent the lineopolar envelope of the same line, with
respect to a syzygetic cubic
a^ + y^ + z^{6Vxyz=:i),
by
(a\ h\ c\ /, g\ hj^x, y, zf = 0,
c. II. 51
402
A MEMOIR ON CUEVBS OF THE THIRD ORDER.
[146
then we have
a' (6c /*) + b' (ca  g*) + c' {ab  h*) + 2f(gh of) + 2g' {hf hg) + 2h' (fg  ch)
{l" + 2l*)(^ + r^ + ^y
+ (21' + 4i  S21H' + 81*) (p + ff + n f7?
+ (•24M'» + mH'^  72W' + 24P + 3) }^^,
which may be verified by writing V = 1, in which case the righthand side becomes
1 + 2P
it should do, 3{PUy. If I' = «/«" » *^** ^' ^^ ^^^ syzygetic cubic be the Hessiib
then the formula becomes
a'{bcf) + &c.=
361'
(l + 4i» + 76Z«)(f + V + f»)» ^
+ 12f' (  1 + 26^' + 56P) (f + .;' + f») f»7?
+ 121 (2 + 57^ + 168i' + 16^") fV(r* J
>■
which is equal to
afeK^*^^^'
26. The equation
{bc' + h'c2ff,...gh'+g'hafa'f,...1t 17, O' =
is the equation in line coordinates of a conic, the envelope of the line which cut<i^
harmonically the conies
(a, b, c, /, g, h $a:, y, zf = 0,
(a', 6', c', /. gf, h'\x, t/, zy = 0;
and if a, b, &c., a', &c. have the values before given to them, then the coefficients
of the equation are
be' + b'c2ff = f {  f + 4«' (Z + I') iff + ^) + iUW  21'  21'*) fv?,
ca' + e'a2gg' =i» {ij» + 4«' (Z + (?* + ?) + (16«'2P 2Z'«)fi,?,
ab' + a'b 2hh' = f (  f»+ 4M' (Z + (? + '?*) +(16«' 2P 2^ ^^
gh' + g'h af  a'f= f {(Z" + /'•) (f> + ,» + f») + {21 + 21' + 8W'>) fij?) + (1 + 4tf' (Z + O) ly^.
A/' + h'f bg' b'g=fi {(P + 1'*) (p + ,^ + (:») + (2Z + 2^ + 8W) fijf} + (1 + 4tt' (f + f)) f»f».
/5^ +/i7  cA'  c'A = r {(P + Z'') (P + 1;* + f) + (2Z + 2Z' + 8W'«) f^f } + (1 + 4«' (f + Z')) f»,» :
and we thence obtain
(bc' + b'c~2/f,..,gh'+g'hafa'/,..Jil r,, ?)• =
+ ( f+ 1'*+ 16U') (? + ff+^)^^
+ (6l + 6l' l24W»)^f»
+ (4 +16(W' + «'*))(,»?'+ f«f»+fS,'). =0
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 403
{is the condition which expresses that a line fa? + lyy + f^ = cuts harmonically its
lineopolar envelopes with respect to the cubic and with respect to a syzygetic cubic.
27. To find the locus of a point such that its second or line polar with respect
to the cubic may be a tangent of the Pippian. Let the coordinates of the point be
(a?, y, z) ; then if fa? + lyy + fer = be the equation of the polar, we have
and the line in question being a tangent to the Pippian,
But the preceding values give
P + ^' + r' = («* + y' + '^)' + 6/(ar» + y» + ^)a?y2'f SGPaj^y^^' f (  2 + 8P) (y*^ + ^ar» f ay )
fiyf = 4fP {aj" \ y^ ■} z^) xyz + (1 + Sl^) /c'y^z^ ^ 21 (y^z' + r^ar'f ay);
and we have therefore
/ (a?" + y» + ?»)' + (10i«  1 6^) (a;* + y' + ^') a:y2: f (1+ 40?  32P) a^fz'' = ;
or introducing IT, HJJ in place of a:* + y* + 2:^, xyz, the equation becomes
which is the equation of the locus in question.
28. The locus of a point such that its second or line polar with respect to the
cubic is a tangent of the Quippian, is found in like manner by substituting the last
mentioned values of f , 17, f in the equation
Q[7 = (l10ZO(P + i;*+H6Z«(5 + 4/*)fi75'.
We find as the equation of the locus,
(1  10i») (a;» + y» + e»)> + 6i (1  30f»  1 6/«) (a:* h y* + ^) ay^ + 6i* (1  104Z'  32/«) ji^y'z'
2(11 8P)» (yV + z^a^ + a^f) = 0,
where the function on the lefthand side is the octicovariant ^„U of my Third
Memoir, the covariant having been in fact defined so as to satisfy the condition in
question. And I have given in the memoir the following expression for 0„jr, viz.
e„ir=(l16f6/«)f^*
+ (6i )U .HU
+ (6i' ){HUf
2(1+ 8P)» {f2^ + «»a;» + aft/').
51—2
404 A MEMOIR ON CURVES OF THE THIRD ORDER. [146
Article Nos. 29 to 31. — Formulod for the intersection of a ciMc curve and a line.
29. If the line ^x^rjy \ ^z = meet the cubic
ic» + y»^ f 6lxyz =
in the points
then we have
It will be convenient to represent the equation of the cubic by the abbreviate^
notation (1, 1, 1, l^x, y, 2^)^ = 0; we have the two equations
(1, 1, 1, Z$^, y, ^)»=0,
farfiyy + f^ =0;
and if to these we join a linear equation with arbitrary coefficients,
aa } /3y \ yz = 0,
then the second and third equations give
and substituting these values in the first equation, we obtain the resultant of the
system. But this resultant will also be obtained by substituting, in the third equation,
a system of simultaneous roots of the first and second equations, and equating to
zero the product of the functions so obtained ^ We must have therefore
(1, 1, 1, ?$^?7i7, 7f< a^  /3f )' = (a^ + ^yi + 7^i) («^2 + ^ya + T^j) (o^j + ^^3 + 7^s) ;
and equating the coefficients of a', I3\ 7*, we obtain the abovementioned relations.
30. If a tangent to the cubic
a^\ y^\z^ \ 6lxyz =
at a point {x^, y^, z^) of the cubic meet the cubic in the point (^,, y,, ^), then
«s : ys : 2^8 = a^ (yi»  z^*) : y^ (V  x^^) : z, (x^^  yi>).
For if the equation of the tangent is f^ + lyy + f^ = 0, then
and
^ : V ' S'=^i* + 2Zyi^, : yi^\2lziXi : z^^ + 2lxiyi.
^ This is in fact the general process of elimination given in Sohlafli's Memoir, '* Ueber die Besultante
einer Systemes mehrerer algebraischer Gleichongen," Vienna Trans. 1S52. [But the prooess was employed much
earlier, by Poisson.]
146] A MEMOm ON CURVES OF THE THIRD ORDER. 405
These values give
= (yi'  ^i') X  (1 + 8P) x,\
since (a^, yi, ^i) is a point of the cubic; and forming in like manner the values of
f* — f and f — rj\ we obtain the theorem.
31. The preceding values of (a?„ y„ z^) ought to satisfy
(a^» + 2/y,2ri) a^ + (yi» f 2h,x,) y, + (^i* + 2lx^i) z^ = 0,
^' + yj' + ^i + 6te,ys^, = ;
in fact the first equation is satisfied identically, and for the second equation we
obtain
x^^ + y.' + ^3' = ^a' (y 1'  ZiJ + y,' (^,'  x,J + ^,» (a:,«  y,J
=  ^1' (y,'  zi')  yi' (V  ^1')  V C^i'  yiO
= (^i' + yi' + ^,') (yi'  z,^) (z,'  ^j») (a:i»  y,%
and consequently
x/ + y»' + z,' + 6/a?,y,2r, = (a^» + y,» + ^j» + 6lx^iZ,) (y,^  z,^) ( ^  x,^) {x,^  y,') = 0.
which verifies the theorem. It is proper to add (the remark was made to me by
Professor Sylvester) that the foregoing values
^3 : ys : ^j = a?, (yi»  z,*) : y, (z,^  a?,') : z^ (x,^  y,')
satisfy identically the relation
a?3^ + ys^ H ^3' _ ^jM^yijff 2^
x^ysZa ^\y\Z\
Article Nos. 32 to 34. — Formulce foi* the Satellite line and point.
32. The line fa? + lyy + 5^ = meets the cubic
a^^y9^2^^ Qlxyz =
in three points, and the tangents to the cubic at these points meet the cubic in
three points lying in a line, which has been called the Satellite line of the given line.
To find the equation of the satellite line; suppose that (xi, y^, z^), (a?j, y,, z^),
(a^s, y3» Z3) are the coordinates of the point in which the given line meets the cubic;
then we have, as before,
(1, 1, 1, «5^?7i;, 7faf, ai?  /3f)« = (oa^ + /9y, + 7^i)(flw?a 4^ /3y2 + 7^a)(<MJ8 + y9y, + 7^3).
406
A MEMOIR ON CURVES OF THE THIRD ORDER.
[146
The equation of the three tangents is
n = [(x,^ + 2ly,z,) X + (yi> + 2lz,x,) y + {z^^ + 2/a?iy0 z'^ =0,
X [(a:,« + 2ly^;) X + (y,« + 2lz^;) y + (V + 2ir^,) 2r]
and if we put
F=(f» + if + ?»)'24?»(p + i7»+f«)fi;?+(24Z48Z*)fV?' + (4 + 32i0(i;'f*+?'f + fV^
(^ is the reciprocant FU of my Third Memoir), then we have identically
and the equation of the satellite line is fa? + 17'^ + 5^2^ = 0. In fact the geometrical
theory shows that we must have
and it is then clear that iV is a mere number. To determine its value in the most
simple manner, write Z = 0, y = 0, a7=f, z = — ^, we have then F, 17—^11 = 0, where
The value of IT is II = F . U, and we thus obtain iV=l. For, substituting the above
values,
rf(aaVai.' + &c.)
+ ?f»(a;,z,'z,' +&C.)
 1*^1 W,
and we have
and thence
and consequently
a^a;,«, + &c. = SJi'f,
(Ci^,?, + &c. =  Sf^,
x,Wz** + &c. = 9?*^ + 6?? » (y  r) = 3?«p + 6(?*i,»,
a;.VV + &c. = 9(7^*  6?»f (f  '?') = 3^?* + ^^W'
n= (7(17* r)"
 r(r'?')'
= (?*?)(?+ 1?* + r  2?''?'  Srp  2pi;').
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 407
Now considering the equation
in order to find f, ri\ f it will be sufficient to find the coefficients of a:*, y", z^ in
the function on the lefthand side of the equation. The coefficient of a^ in 11 is
{a;,^ + 2lyiZi) (irj" + 2lz^^) (a^* + 2Za?,ys)
^ •Crj •C/2 «*^
+ 2? {onxx^y^z + &c.)
+ 4i'» (x^y^^^t + &c.)
• +8Z» yiy^i;siz^t\
and it is easy to see that representing the function
(1, 1, 1, /$)8? 717, 7f  a?, ai7  fi^f
by
(a, b, c, f, g, h, i, j, k, l$a, /8, 7)»,
the symmetrical functions can be expressed in terms of the quantities a, b, &c., and
that the preceding value of the coefficient of a^ in H is
a'
f 21 (9hj  6al)
+ 4Z» (6gk  3fj  Shi + 31»)
+ 8Z' be;
and substituting for a, &c. their values, this becomes
+ 4^' { 6 (?r + 21^) (pi, + 2Zf C)
+ 3(i,{:» + 2f?p)((7f + 2i5i;«)
+ 3 (?.;' + 2?i,r') (»?•?+ 2^P))
and reducing, we obtain for the coefiScient of a* in 11 the following expression,
18/ ^*^
 24? (p + 1;" + {:») fi^r
24Z»(i7»i;»+C»» + f.7»)
408
A M£MOIR ON CURVES OF THE THIRD ORDER.
[146
Now the coefficient of a^ in F , U is simply F, which is equal to
f«+ ^«+ f» 2i;»?» 2f*f  2f i7»
and subtracting, the coefficient ofa^ in F. U —Tl is
f  2f i;»  2f ?»
 48^f^^?»,
which is equal to
(1 + 8P) f (f*  2fi;»  2f ?»  6lvV)'
The expression last written down is therefore the value of pf, or dividing by ^ we
have f, and then the values of rj', f are of course known, and we obtain the
identical equation
F. U^U =
(l + 8P)(f^+W+?^)' ^
+ (17*
+ (?*
2^7»  2^f»
2i7?»2i;p
2rr  2?i;»
6zrr)y
and the second factor equated to zero is the equation of the satellite line of
fa? + i;y + fg: = 0.
33. The point of intersection of the line f a? + lyy + f^ = with the satellite line
^'x + i;'y + f ^ = is the satellite point of the former line ; and the coordinates of the
satellite point are at once found to be
: (PV)(f^ + 2in
34. If the primary line ^x + rfy + fy = is a tangent to the cubic, then (a?,, y,, ^i)
being the coordinates of the point of contact, we have
^ : 1) ■ ?=«i' + 2/yi^, : yi*\r'ilz^ : «i*+2^,;
146]
A MEMOIR ON CURVES OF THE THIRD ORDER.
409
these values give as before
and they give also
and consequently we obtain
that is, the satellite point of a tangent of the cubic is the point in which this
tangent again meets the cubic.
Article Nos. 35 and 36. — Theorems relating to the satellite point.
35. If the line f a? 4 i;y + ?2r = be a tangent of the Pippian, then the locus of
the satellite point is the Hessian.
Take (x, y, z) as the coordinates of Jbhe satellite point, then we have
x:y : ^ = (i;»  ?») (17?+ 2/p)
:(CP)(Sf+2Zi;»)
where the parameters f, 1;, 1^ are connected by the equation
i(?+i;'+C')+(i+4P)fi;r=0.
We have
&ad it is easy to see that the function on the righthand side must divide by tf — i^:
hence a^ + y' + s^ will also divide by 17* — f*, and consequently by (ij*  {?) (S* — ?) (? — «?*)•
We have
I + ?
+ 6?fV?* {  ^  rV  1/' + 3p (?» + 1;»)  3f )
+ 12?^,? {rf^(rf+^) + 3p,^C  f }
+ 8i» {  i/'f + 3i;»r?  (V + D PI
and
C. II.
52
410 A MEMOIR ON CURVES OF THE THIRD ORDER. [146
Adding these values and completing the reduction, we find
(*■» + y* + z») ^ (»?' f) (?'?) (f»  »7») =  ^  1?« ?•+ Zi/T + 2{:»? + 2f»7'
+ 18i ^*^
+ 12P(' + i7' + f»)fi,?
+ 8i» (i/'C + Cf + fi?») ;
and we have also
+ 8Z' f«v'r,
and thence
A i^ + ri' + ^y
+ (UPA + ilB) (f* + 17» + f») fi^r
+ (18^4 + (1 + 8?) 5) pi,«i;»
+ ((4P + 81*) A + 4P5) (vT + rf + f 'j')
The coefficient of ij«?» + f »f» + f <;» on the righthand side will vanish if (1 + 2?) 4 + i* 5 = 0,
or, what is the same thing, if A=l^, 5 = — (1 + 2Z'); and substituting these values, we
obtain
{P {x> + y' + z>)(l + 2P) f,C} H (i;*  ?») (C  p) (p  ,;»)
= P (f + 'T' + H
+ ( 4Z + 4i*) (t' + 1?' + f) fijf
+ ( 1 + 8P  lei*) f*ij»?»,
or, what is the same thing,
P (a;* + y" + ^)  (1 + 2P) a;y^ =  (i,»  H (r  f ) (? • 7*)
x[li^ + v'+^) + (l + U*) f.7?}'.
Hence the lefthand side vanishes in virtue of the relation between ^, rj, ^, or we have
^{x> + fiz'){l + 2P)xyz = 0.
which proves the theorem.
36. Suppose that {X, Y, Z) are the coordinates of a point of the Hessian, and
let (P, Q, R) be the coordinates of the point in which the tangent to the Hessian
at the point {X, Y, Z) again meets the Hessian, or, what is the same thing, the
146]
A MEMOIB ON CURVES OF THE THIRD ORDER.
411
satellite point in regard to the Hessian of the tangent at (X, F, Z), And consider
the conic
X{x'^' 2lyz) + Tif^^ 2lzx) + Z(a^'{ 2lxy\
which is the first or conic polar of the point (X, F, Z) in respect of the cubic. The
polar (in respect to this conic) of the point (P, Q, R) will be
w
here
f = PZ + /(i2F+(2Z),
i=RZ + l(QX+PY);
or putting for (P, Q, iJ) their values,
^ = (Y''Z')(X*IYZ),
V = (Z'X')(7^^1ZX),
r = (Z»P)(^iZF);
and if from these equations and the equation of the Hessian we eliminate {X, F, Z)
we shall obtain the equation in line coordinates of the curve which is the envelope
of the line fa? + i;y + 5^ = 0. We find, in fact,
p + ,7»+ r = (F«  iP'XZ' Z») (Z»  F«)
SI {X'+T' + Z*)XYZ
+ 91* X^Y^Z^
(,+ (! W){Y^Z* + ^Z» + Z^F'),
X ^
^?
= (F'^)(^  X»)(X» P)
r
J
Z'(y*+F' + ^)ZF^
i (7'^ + ^Z» + Z»F');
and thence recollecting that
HU^J? (Z» + F» + Z»)  (1 + 2P) ZF^,
we find
and the equation of the envelope is
which is therefore the Pippian. We have thus the theorem:
52—2
412 A MEMOIR ON CURVES OF THE THIRD ORDER. [146
The envelope of the polar of the satellite point in respect to the Heesian of the
tangent at any point of the Hessian, such polar being in respect of the conic which
is the first or conic polar of the point of the Hessian in respect of the cubic, is the
Pippian.
Article Nos. 37 to 40. — Investigations and theorems relating to the first or conic polar
of a point of the cubic.
37. The investigations next following depend on the identical equations
[a(X^ + 2lYZ){l3{Y'^2lZX)^y(Z^ + 2lXY)]
= {Z (iB* + 2lyz) + F (y« + 2lzx) + Z («» + 2%)}
h {x (Z« + 21YZ) + y (F» + 21ZX) '{z{Z'\ 21XY)\
x{(aF^fi9ZZf7ZF)(Za;»+Fy» + Z^«) + (aZ« + i9F» + 7Z*)(ZF?+Fzx + Za;y)l.
which is easily verified.
I represent the equation in question by
then considering (a?, y, z) as current coordinates, and (Z, Y, Z) and (a, /8, 7) as the
coordinates of two given points 2 and fl, we shall have 17=0 the equation of the
cubic, TT = the equation of the first or conic polar of S with respect to the cubic,
P = the equation of the second or line polar of 2 with respect to the cubic The
equation T = is that of a syzygetic cubic passing through the point 2 : the
coordinates of the satellite point in respect to this syzygetic cubic of its tangent at
2 are
XiY^'Z') : Y{Z^^X^) : Z(Z» F»);
and calling the point in question 2^ then Z = is the equation of a line through
the points 2', ft. The equation = is that of a conic, viz. the first or conic polar
of 2 with respect to a certain sjrzygetic cubic
 2 {aYZ^^ZX + 7ZF) («» + y» + ^) + (aZ» + /8F» + 7^)ajyir = 0,
depending on the points 2, ft, or, what is the same thing, the conic B = is a
properly selected conic passing through the points of intersection of the first or conic
polars of 2 with respect to any two sjrzygetic cubics; and lastly, .E' is a constiint
coefficient. The equation expresses that the points of intersection of
(ir=o, p = o), (Tr=o, e = o), (Z = o, p=o), (Z = o, e = o),
lie in the syzygetic cubic T = 0.
146] A MEMOIB ON CURVES OF THE THIRD OKDEB. 413
The lefthand side of the equation may be written
 ZF^ {a (Z« + 2ZFZ) + /8 ( F« + 2/ZZ) + 7 (^ + 2iZF)l(rc» + y« + ^ + eZfljy^)
+ xyz{a(X^ + 2l7Z)+fi(7^ + 2lZX) + y{Z» + 2lXY)}(X^^Y^ + Z' + 6lXYZ);
and it may be remarked also that we have
3ZFZ{a(Z» + 2iFZ) + /9(F« + 2ZZZ) + 7(^ + 2/ZF)}
equal identically to
(Z(F»Z»)(7F/8^)+F(Z*.Z»)(a^7Z) + ^(Z«F»)(i8ZaF)}
(aFZ + /SZZ + 7ZF)(Z«+ Y' ^ Z' + eiXYZ).
Hence if we assume
Z»+ F» + Z» + 6?ZFZ=0,
the equation will take the form
where the constant coefficient K may be expressed under the two different forms
if = ZFZ{a(Z« + 2iFZ) + /8(F» + 2/ZZ) + 7(^ + 2ZZF)}
= J {Z(F'  Z*) (7F i8Z)+ F(^ Z«)(aZ 7Z) + Z(Z» Y*){^X  aF)},
and W, Z, P, have the same values as before. In the present case the point S
is a point of the cubic : the equation IT = represents the first or conic polar of
the point in question, and the equation P = its second or line polar, which is also
the tangent of the cubic. The line Z = is a line joining the point XI with the
satellite point of the tangent at S, or dropping altogether the consideration of the
point ft, is an arbitrary line through the satellite point: the first or conic polar of
2 meets the cubic twice in the point 2, and therefore also meets it in four other
points; the conic © = is a conic passing through these four points, and com
pletely determined when the particular position of the line through the satellite
point is given. And, as before remarked, = is a conic passing through the points
of intersection of the first or conic polars of 2 with respect to any two syzygetic
cubics. We have thus the theorem :
The first or conic polar of a point of the cubic touches the cubic at this point,
and besides meets it in four other points; the four points in question are the points
in which the first or conic polar of the given point in respect of the cubic is
iutersected by the first or conic polar of the same point in respect to any syzygetic
cubic whatever.
38. The analytical result may be thus stated: putting
K = ^YZ'^/3ZX + yXY, X = aZ« + /SF» + 7Z«,
414
A MEMOm ON CUBVES OF THE THULD ORDER.
[146
or, if we please, considering k, \ as arbitrary parameters, then the four points lie in
the conic
(2kX, 2/cT, 2/cZ, \Z, \F, XZ$a?, y, 2r)> = 0,
or, what is the same thing, they are the points of intersection of the two conies
Za^ + Fy« + Zz^ = 0,
Xyz + Yzx + Zxy = 0.
39. Considering the four points as the angles of a quadrangle, it may be shown
that the three centres of the quadrangle lie on the cubic. To effect this, assume
that the conic
(2/cZ, 2k7, 2kZ, XZ, \F, \Z\x, y, z>^=^0
represents a pair of lines; these lines will intersect in a point, which is one of the
three centres in question. And taking x, y, z a& the coordinates of this point, we
have
a^ : y^ \ z^ : yz I zx I xy ^ 4/^* YZ — X'Z*
4/e»ZZ X»F»
4/e» XT" X»^
X» YZ + 2k\X^
X»ZZ + 2/eXF«
X«ZF+2/eX^;
and we may, if we please, use these equations to find the relation between /c, X
Thus in the identical equation a^ . y* — (fl?y)" = 0, substituting for a^, xy, y" their values,
and throwing out the factor Z, we find (4#c>X»)ZFZ/eX«(Z«+ F« + ^) = 0, and
thence, in virtue of the equation Z'+ F* + ^ + 6?ZFZ = 0, we obtain
4#c> f 6?/eX»  X» = 0.
But the preceding system gives conversely,
X* : Y* : Z" : YZ : ZX : XY^^i^z XV
^i^zx  xy
4/c'a?y — XV
Xh/z + 2/cXflj«
X^zx + 2/cXy*
X^xy + 2/eX«».
Hence firom the identical relation Z*. F»(ZF)» = 0, substituting for X\ XY, P
their values, and throwing out the £sbctor z, we find (4ie* — X')a5y2r — /eX*(«* + y' + z*) = 0,
and thence, in virtue of the equation 4#c* — X' = — 6i/icX*, we obtain
^ + y* + ^ + Qlxyz = 0,
146] A MEMOIB ON CURVES OF THE THIRD ORDER. 415
which shows that the point in question lies on the cubic. We have thus the
theorem :
The first or conic polar of a point of the cubic touches the cubic at the point,
and meets it besides in four points, which are the angles of a quadrangle the
centres of which lie on the cubic. In other words, the quadrangle is an inscribed
quadrangle.
40. To find the equations of the three axes of the quadrangle, that is of the
lines through two centres.
We have
(4/e» 7Z  \^X') x + ( \'XY^ ^KkZ") y + ( X»^Z + 2/eX7«) ^ = 0,
( \^XY{ 2k\Z^) X + (4^ZX  X«r») y + ( X'FZ + 2ic\Z«) ^ = 0,
( X'ZZ + 2/rXF>) a? + ( X«7Z + 2/rXZ»)y+(4/e»ZF X«Z0r=O;
or arranging these equations in the proper form and eliminating ic", k\ X', we find
YZx, Z^\Y^z, X{Xx^Yy\Zz) =0;
ZXy, XH^Z'x , Y{ XxYy^ Zz)
XYz, r»a? + Z«y, Z{ Xx \ Yy \ Zz)
or, multiplying out,
ZFZ{(^^F»)a;» + (Z»^)y» + (r»Z»)»'}
f a^yZY^ ( 2Z» + F» + ^) + zai'YZ^ (2X^ Y^^Z*)
+ y'zXZ^ ( 2 F» + ^ f Z») + xfZX* (2F»  2>  Z»)
+ z'xYX*( 2Z* + Z» + F») + y^»ZF'(2^  Z*  F») = 0.
We may simplify this result by means of the equation Z»+ F» + ^ + 6ZZFZ= 0, so as
to make the lefthand side divide out by XYZ: we thus obtain
(^F»)a:» + (Z»i?')y» + (F»Z»)?'
+ ( 3Z»F. 6iF>Z)^ + ( 3F»Z 6lZ'X)y*z + ( 3^Z  6lX'Y)^x
+ ( 3ZF«+6ZZ>Z)«y« + ( 3FZ*+6/F»Z)y^»( 3^Z^ + 6/^F)^a^ = ();
or in a difierent form,
+ ( Sx'y  eiz^x) X^Y+( 3y'z  Gla^^y) Y'Z + ( 3z'x  6ly^z) Z'X
+ ( fixy''h6lyz^)XY^ + ( Syz^ + 6lzaf) YZ' + ( Szx' \ Qlxf) ZX^ == 0,
as the equation of the three axes of the quadrangle.
416 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46
Article No. 41. RecapitulcUum of geometrical definxtione of the Pippian.
In conclusion, I will recapitulate the different modes of generation or geometrical
definitions of the Pippian, obtained in the course of the present memoir. The curve
in question is:
1. The envelope of the line joining a pair of conjugate poles of the cubic (see
Nos. 2 and 13).
2. The envelope of each line of the pair forming the first or conic polar with
respect to the cubic of a conjugate pole of the cubic (see Nos. 2 and 14).
3. The envelope of a line which is the polar of a conjugate pole of the cubic,
with respect to the conic which is the first or conic polar of the other conjugate pole
in respect to any syzygetic cubic (see Noa 2 and 9).
4. The locus of the harmonic with respect to a pair of conjugate poles of the
cubic of the third point of intersection with the Hessian of the line joining the two
conjugate poles (see Nos. 2 and 17).
6. The envelope of a line such that its lineopolar envelope with respect to the
cubic breaks up into a pair of lines (see No. 24).
6. The envelope of a line which meets three conies, the first or conic polars of
any three points in respect to the cubic, in six points in involution (see No. 22^
7. The envelope of the second or line polar with respect to the cubic, of a point
the locus of which is a certain curve of the sixth order in quadratic syzygy with
the cubic and Hessian, viz. the curve — S . IT"* + (fiVy = (see No. 27).
8. The envelope of a line having for its satellite point a point of the Hessian
(see No. 35).
9. The envelope of the polar of the satellite point with respect to the Hessian
of the tangent at a point of the Hessian, with respect to the first or conic polar of
the point of the Hessian in respect to the cubic (see Na 86).
147.
I
I
A MEMOIR ON THE SYMMETRIC FUNCTIONS OF THE ROOTS
OF AN EQUATION.
[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for
the year 1857, pp. 489 — 499. Received December 18, 1856, — Read January 8, 1857.]
There are contained in a work, which is not, I think, so generally known as it
deserves to be, the "Algebra" of Meyer Hirsch [the work referred to is entitled
Sammlung von Beispielen Formeln und Aufgaben aus der Buchstabenrechnung und
Algebra, 8vo. Berlin, 1804 (8 ed. 1853), English translation by Ross, 8vo. London,
1827] some very useful tables of the symmetric functions up to the tenth degree
of the roots of an equation of any order. It seems desirable to join to these a set of
tables, giving reciprocally the expressions of the powers and products of the coefficients
in terms of the symmetric functions of the roots. The present memoir contains the
two sets of tables, viz. the new tables distinguished by the letter (a), and the tables
of Meyer Hirsch distinguished by the letter (6) ; the memoir contains also some
remarks as to the mode of calculation of the new tables, and also as to a peculiar
symmetiy of the numbers in the tables of each set, a synmietry which, so far as I
am aware, has not hitherto been observed, and the existence of which appears to
constitute an important theorem in the subject. The theorem in question might, I
think, be deduced from a very elegant formula of M. Borchardt (referred to in the
sequel), which gives the generating function of any sjonmetric function of the roots,
and contains potentially a method for the calculation of the Tables (6), but which,
fix)m the example I have given, would not appear to be a very convenient one for
actual calculation.
Suppose in general
(1, 6, C...5I, a?)* =(1 — 6a?)(l— )8a:)(l — 7a:)... ,
so that
— 6 = 2a, + c = 2a)8, — d = %afiy, &c.,
and if in general
0. n. 53
418
A MEMOIB ON THE SYMMETRIC FUNCTIONS
[147
where as usual the summation extends only to the distinct terms, so that e.g. (/?*)
contains only half as many terms as (pq), and so in all similar cases, then we have
6 = (l), +c = (P), d = (P), &c.;
and the two problems which arise are, first to express any combination fr'^c^... in terms
of the symmetric functions (l^m^..,), and secondly, or conversely, to express any
symmetric function {l^m^ ...) in terms of the combinations b^d^,...
It will conduce materially to brevity if 1^29... be termed the partition belonging
to the combination b^d^... ; and in like manner if l^m^... be termed the partition
belonging to the symmetric function (l*m^,,,), and if the sum of the component
numbers of the partition is termed the weight.
Consider now a line of combinations corresponding to a given weight, e.g. the
weight 4, this will be
e bd d" b^c b" (line)
4 13 2» 1«2 1*,
where I have written under each combination the partition which belongs to it, and
in like manner a colunm of symmetric functions of the same weight, viz.
(4) (column)
(31)
(2»)
(21*)
(1*).
where, as the partitions are obtained by simply omitting the ( ), I have not separately
written down the partitions.
It is at once obvious that the different combinations of the line will be made up
of numerical multiples of the symmetric functions of the column ; and conversely, that
the symmetric functions of the column will be made up of numerical multiples of the
combinations of ^ the line ; but this requires a further examination. There are certain
restrictions as to the symmetric functions which enter into the expression of the com
bination, and conversely, as to the combinations which enter into the expression of the
symmetric function. The nature of the first restriction is most clearly seen by the
following Table:
Number of
Greatest
Parts.
Part.
1
4
2
3
2
2
3
2
4
1
Combinations
with their several
Partitions.
6
bd
b^o
b*
4
13
2«
1«2
1*
Contain Maltiples of the
Symmetrio Funotions.
(1*),
(21»),
(21'),
(2'),
(2^), (31),
(2'), (31), (4)
Greatest Part
does not exceed
1
2
2
3
4
Number of j
Parts not
less than
4
3
2
2
1
147] OF THE ROOTS OF AN EQUATION. 419
Thus, for instance, the combination bd (the partition whereof is 13) contains multiples
of the two symmetric functions (1*), (21*) only. The number of parts in the partition
13 is 2, and the greatest part is 3. And in the partitions (1*), (21') the greatest part
is 2, and the number of parts is not less than 3. The reason is obvious : each term of
the developed expression of bd must contain at least as many roots as are contained
in each term of d, that is 3 roots, and since the coefficients are linear functions in
respect to each root, the combination bd cannot contain a power higher than 2 of any
root. The reasoning is immediately applied to any other case, and we obtain
First Restriction. — A combination b^c^... contains only those symmetric functions
(l*mv.,,), for which the greatest part does not exceed the number of parts in the
partition P2^... , and the number of parts is not less than the greatest part in the
same partition.
Consider a partition such as 1*2, then replacing each number by a line of units
thus,
1
1
11,
and summing the columns, we obtain a new partition 31, which may be called the
conjugate^ of 1*2. It is easy to see that the expression for the combination 6*c (for
which the partition is 1'2) contains with the coefficient unity, the symmetric function
(31). the partition whereof is the conjugate of 1*2. In fact 6"c = (— 2a)" (Sa/8), which
obviously contains the term + lo*^, and therefore the symmetric function with its
coefficient + 1 (31) ; and the reasoning is general, or
Theorem. A combination 6^c^... contains the sjrmmetric function (partition conjugate
to 1''2^...) with the coefficient unity, and sign + or — according as the weight is even
or odd.
Imagine the partitions arranged as in the preceding column, viz. first the partition
into one part, then the partitions into two parts, then the partitions into three parts,
and so on; the partitions into the same number of parts being arranged according to
the magnitude of the greatest part (the greatest magnitude first), and in case of
equality according to the magnitudes of the next greatest part, and so on (for other
examples, see the outside column of any one of the Tables). The order being thus
completely defined, we may speak of a partition as being prior or posterior to another.
We are now able to state a second restriction as follows.
Second Restriction. — The combination b^cfi.,. contains only those symmetric functions
which are of the form (partition not prior to the conjugate of 1*2*...).
The terms excluded by the two restrictions are many of them the same, and it
might at first sight appear as if the two restrictions were identical; but this is not
^ The notion of Conjugate Partitions is, I believe, due to Professor Sylvester or Bfr Ferrers. [It was dne to
Mr now Dr Ferrers.]
53—2
420
A MEMOIR ON THE SYMMETRIC FUNCTIONS
[147
so : for instance, for the combination bd\ see Table VII (a), the term (41*) is excluded
by the first restriction, but not by the second; and on the other hand, the term
(3*1), which is not excluded by the first restriction, is excluded by the second restriction,
as containing a partition 3^1 prior in order to 32', which is the partition conjugate
to 13', the partition of bd\ It is easy to see why b(P does not contain the symmetric
function (3'1); in fact, a term of (3*1) is (a'/3'7), which is obviously not a term of
6d' = (— 2a) (Sa^y)* ; but I have not investigated the general proof
I proceed to explain the construction of the Tables (a). The outside column
contains the symmetric fimctions arranged in the order before explained; the outside or
top line contains the combinations of the same weight arranged as follows, viz. the
partitions taken in order from right to left are respectively conjugate to the partitions
in the outside column, taken in order from top to bottom ; ia other words, each square
of the sinister diagonal corresponds to two partitions which are conjugate to each other.
It is to be noticed that the combinations taken in order, from left to right, are not
in the order in which they would be obtained by Arbogast's Method of Derivations
from an operand a*, a being ultimately replaced by unity. The squares above the
sinister diagonal are empty (i.e. the coefficients are zero), the greater part of them in
virtue of both restrictions, and the remainder in virtue of the second restriction; the
empty squares below the sinister diagonal are empty in virtue of the second restriction;
but the property was not assumed in the calculation.
The greater part of the numbers in the Tables (a) were calculated, those of each
table fi"om the numbers in the next preceding table by the following method,
depending on the derivation of the expression for b^^^c^... from the expression for b'^c^...
Suppose, for example, the column cd of Table V(a) is known, and we wish to calculate
the column bed of Table VI (a). The process is as follows :
Given
we obtain
2n 21V 1»
3 10
321
2'
31»
2n«
21*
1«
1
3
2
3
6
12
10
60
1
3
3
8
22
60
where the numbers in the last line are the numbers in the column bed of Table
VI (a). The partition 2*1, considered as containing a part zero, gives, when the parts
are successively increased by 1, the partitions 321, 2', 2*1', in which the indices of the
increased part (i.e. the original part plus unity) are 1, 3, 2; these numbers are taken
as multipliers of the coefficient I of the partition 2'1, and we thus have the new
coefficients 1, 3, 2 of the partitions 321, 2*, 2*1*. In like manner the coefficient 3 of
147] OF THE ROOTS OF AN EQUATION. 421
the partition 21' gives the new coefficients 3, 6, 12 of the partitions 31', 2*1*, 21*,
and the coefficient 10 of the partition 1* gives the new coefficients 10, 60 of the
partitions 21* and 1", and finally, the last line is obtained by addition. The process
in fact amounts to the multiplication separately of each term of cd =
1 (2»1) + 3 (21') + 10 (1»)
by 6 = (1). It would perhaps have been proper to employ an analogous rule for the
calculation of the combinations M^.,. not containing 6, but instead of doing so I
availed myself of the existing Tables (6). But the comparison of the last line of each
Table (a) (which as corresponding to a combination b^ was always calculated in
dependently of the Tables (6)) with such last line as calculated from the corresponding
Table (6), seems to afford a complete verification of both the Tables ; and my process
has in fact enabled me to detect several numerical errors in the Tables (6), as given
in the English translation of the work above referred to. It is not desirable, as
regards facility of calculation and independently of the want of verification, to calculate
either set of Tables wholly from the other; the rules for the independent calculation
of the Tables (6) are fully and clearly explained in the work referred to, and I have
nothing to add upon this subject.
The relation of symmetry, alluded to in the introductory paragraph of the present
memoir, exists in each Table of either set, and is as follows: viz. the number in the
Table corresponding to any two partitions in the outside column and the outside line
respectively, is equal to the number corresponding to the same two partitions in the
outside line and the outside column respectively. Or, calling the two partitions P, Q,
and writing for shortness, combination (P) for the combination represented by the
partition P, and for greater clearness, symmetric function (P) (instead of merely (P))
to denote the symmetric function represented by the partition P, we have the following
two theorems, viz.
Theorem. The coefficient in combination (P) of symmetric function (Q) is equal
to the coefficient in combination (Q) of symmetric function (P);
and conversely.
Theorem. The coefficient in symmetric function (P) of combination (Q) is equal
to the coefficient in symmetric function (Q) of combination (P).
M. Borchardt's formula, before referred to, is given in the *Monatsbericht' of the
Berlin Academy (March 5, 1885) S and may be thus stated; viz. considering the case of
n roots, write
(1, 6, c, ... k"$l, xf = (1  aa;)(l  ^a?)...(l  kx) =/b,
then
1 1 _i ^N ^ 1 / x„ My'f^ A ^ A. n(^,y,...ti)
J ax l^y'"l Ku) k^ ^ Il(Xyy,...u) dx dy" du fxjy...fu '
1 And in Crelle, t. liii. p. 195.— Note added 4th Deo. 1857, A. C.
422 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147
where 11 (x, y,.,,u) denotes the product of the differences of the quantities x, y,..,v,
and on the lefthand side the summation extends to all the different permutations of
a, ^, ... /t, or what is the same thing, of x, y,...u.
Suppose for a moment that there are only two roots, so that
(1, 6, cth xy^(l^ax)(l^l3xl
then the lefthand side is
1 1
which is equal to
2 + (a + i9)(a? + y) + (a» + ^)(a;» + y') + 2a/8ajy + (a» + /80(aJ» + y') + (a»i8 + <^^
and the righthand side is
1 fx/y d d xy
which is equal to
c ' xy dx dy fxfy '
1 My [ r^fyryfaH^y)rxfy \
c x — y
(f^r {fyf
and therefore to
c fxfy { xy J J i^y
or substituting for fx^ fy their values,
f^fy fyfa
xy
becomes equal to
2c  62  6c(a; + y)  2c»icy,
and fxfy is equal to
6» + 2ic (a? H y) + ^xy.
The righthand side is therefore equal to
2 + 6(a? + y) + 2cay .
(l+6a? + cic»)(l + 6y + cy«)'
and comparing with the value of the lefthand side, we see that this expression may
be considered as the generating function of the sjrmmetric functions of (a, /S), viz. the
expression in question is developable in a series of the symmetric functions of (a:, y\
the coefficients being of course functions of 6 and c, and these coefficients are (to
given numerical factors yrls) the symmetric functions of the roots (a, /8).
147]
OP THE ROOTS OP AN EQUATION.
423
And in general it is easy to see that the lefthand side of M. Borchardt's
formula is equal to
[0] + [1] (1) (ly + [2] (2) (2)' + [P] (1») (1')' + &c,
where (1), (2), (1'), &c. are the symmetric functions of the roots (a, A ••• fc)y (1)', (2)',
{V)\ &c. are the corresponding symmetric functions of («, y, ...w), and [0], [1], [2], [1*],
&c. are mere numerical coefficients; viz. [0] is equal to 1.2.3...n, and [1], [2], [1'], &c.
are such that the product of one of these factors into the number of terms in the
corresponding sjrmmetric function (1), (2), (P), &c. may be equal to 1.2.3...n. The
righthand side of M. Borchardt's formula is therefore, as in the particular case, the
generating function of the symmetric functions of the roots (a, ^, ... ^), and if a
convenient expression of such righthand side could be obtained, we might by means
of it express all the sjrmmetric functions of the roots in terms of the coefficienta
Tahles rdating to the Symmetric Functions of the Roots of an EquaMon.
The outside line of letters contains the combinations (powers and products) of the
coefficients, the coefficients being all with the positive sign, and the coefficient of the
highest power being unity; thus in the case of a cubic equation the equation is
a;^ + 6«* + w? + d = (a?  a) (a?  )8) (a?  7) = 0.
The outside line of numbers is obtained from that of letters merely by writing 1, 2, 3...
for 6, c, d..., and may be considered simply as a different notation for the combinations.
The outside column contains the different symmetric functions in the notation above
explained, viz. (1) denotes So, (2) denotes So", (1') denotes SayS, and so on. The Tables
(a) are to be read according to the columns; thus Table 11(a) means 6" = 1 (2) + 2(1)*,
c = (l*). The Tables (6) are to be read according to the lines; thus Table 11(6)
means (2) =  2c + 1ft*, (1») = + Ic.
1(a).
(1)11
1(6).
(1)
1
6
 1
II (a).
(2)
(I'O
2
c
+ 1
1»
+ 1
+ 2
II (6).
2
c
P
6*
(2)
(1')
m(o).
3
12
1»
11
d
he
6»
(3)
 1
(21)
 1
3
(1')
1
3
6
m (6).
3 12
1»
d he
6»
(3)
31 + 3
 1
(21)
+ 31
(1»)
 1
424
A MEMOIR ON THE SYMMETRIC FUNCTIONS
[147
IV (a).
(5)
(41)
(32)
(31«)
(2'1)
(2P)
(!•)
II
(4)
(31)
(2«)
(21')
(1^)
4
13
2^
P2
V
e
bd
c*
b^c
b*
+ 1
+ 1
+ 1
+ 2
+ 4
+ 6
+ 1
+ 1
+ 4
+ 2
+ 6
+ 5
+ 12
+ 12
+ 24
V(o).
5
/
 i
14
be
 1
5
23
cd
P3
b*d
12*
6c«
P2
b^c
b'
1
 3
1
 5
1
 2
 10
 1
 7
 12
27
 20
 1
 2
 7
 5
 30
 60
 3
10
12
20
30
60! 120
IV
(b).
i 13
2«
1«2
I*
^
«
bd
c«
ft»c
6*
(4)
4
+ 4
+ 2
 4
+ 1
(31)
+ 4
 1
2
+ 1
(2')
+ 2
2
+ 1
(21')
4
+ 1
(1')
+ 1
V
(6).
5
14
23
P3
12«
P2
P
=
/
be
cd
b^d
b<^
6»c
6»
(5)
5
+ 5
+ 5
5
5
+ 5
1
(41)
+ 5
 1
5
+ 1
+ 3
 1
(32)
+ 5
5
+ 1
+ 2
 1
(31»)
5
+ 1
+ 2
 1
(2*1)
 5
+ 3
 1
(2P)
+ 5
 1
(P)
 1
1
II
6
9
15
¥
+ I
24
ce
P4
b^e
3»
d"
VI (a]
123
bed
1.
P3
b^d
2»
P2«
P2
b'c
6«
(6)
i
+ 1
(51)
+ 1
+ 6
(42)
+ l!+ 4
+ 15
(3')
+ 1
+ 2
+ 6
+ 20
(41')
+ 1
• • •
+ 2
+ 9
+ 30
(321)
+ 1
+ 3
+ 3
+ 6
+ 8
+ 22
+ 60
(2')
+ 1
+ 3
+ 6
+ 15
+ 36
+ 90
(31')
+ 1
• • •
+ 3
+ 10
+ 6
+ 18
+ 48
+ 120
(2»1')
+ 1
+ 2
+ 2
+ 8
+ 18
+ 15
+ 34
+ 78
+ 180
(21*)
+ 4
+ 15
+ 9
+ 6
+ 22
+ 48
+ 36
+ 78
+ 168
+ 360
(1)
+ 1
+ 6
+ 30
+ 20
+ 60
+ 120
+ 90
+ 180
+ 360
+ 720
VI (6).
(6
(51
(42
(3«
(4P
(321
(2"
(31
(2«P
(2P
6
9
15
¥
+ 6
24
ee
+ 6
P4
6»c
6
+ 1
+ 2
+ 3
 1
3
• • •
+ 1
3«
d*
+ 3
3
3
+ 3
+ 3
3
+ 1
123
bed
P3
b^d
+ 6
 1
 2
• • •
+ 1
2»
c»
 2
+ 2
2
+ 1
P2>
6V
+ 9
4
+ 1
P2
b*c
6
+ 1
1*
+ 1
 6
 12
+ 6
l6
+ 7
+ 6
6
3
+ 1
+ 7
+ 2
 1
4
+ 1
+ 2
3
+ 2
+ 4
2
2
+ 1
+ 4
+ 3
 3
 6
 3
 12
+ 1
 2
+ 6
+ 9
 6
+ 1
147]
OP THE ROOTS OF AN EQUATION.
425
7
h
16
^9
25
1«5
34
de
124
hce
1»4
6»«
VII (
12«
[a),
2«3
c^d
P23
h^cd
P3
h^d
12»
6c»
P2«
6V
P2
6»c
V
 1
7
1
1
7
 21
1
5
 1
3
 10
 35
 42
 1
• • •
2
 11
1
 1
 4
 3
 11
 35
 105
 1
 2
 6
 7
 18
 50
 140
 1
 2
 5
 12
 12
 31
 80
 210
1 1
• • •
• • •
 3
 13
 6
 24
 75
 210
 1
 3
 2
 5
 13
 34
 27
 68
 170
 420
 1
 3
 6
 13
 7
 12
 27
 60
 51
 117
270
 630
 1
• • •
 4
 6
12
 34
 88
 60
 150
 360
 840
 1
 2
 3
 11
 24
 18
 31
 68
 150
117
 25«
 570
 1260
 5
 11
10
35
 35
 75
 50
 80
 170
360
270
 570
 1200
2520
 1
21
42
 105
210
 140
210
420
840
630
 1260
2520
5040
(7)
(61)
(52)
7
h
 7
+ 7
+ 7
16
hg
+ 7
 1
7
7
+ 1
+ 8
+ 4
+ 7
 1
9
5
+ 1
+ 5
 1
25
cf
+ 7
7
+ 3
7
+ 2
+ 4
+ 7
3
 2
6
+ 3
+ 2
 1
1»5
bV
7
+ 1
+ 2
+ 7
 1
3
4
2
+ 1
+ 4
• • •
 1
34
de
+ 7
7
7
+ 5
+ 7
+ 2
5
+ 1
3
+ 3
 1
124
bee
 14
+ 8
+ 4
+ 2
 3
 8
+ 1
+ 2
+ 3
 1
VII
1'4
6»e
+ 7
 1
2
3
+ 1
+ 3
• • •
• • •
 1
(6).
ll«
bd^
7
+ 4
+ 7
5
4
+ 1
+ 2
 1
2«3
cH
7
+ 7
3
+ 1
2
+ 2
 1
1
P23
h'cd
+ 21
 9
 6
+ 3
+ 4
 1
P3
b*d
7
+ 1
+ 2
• • •
 1
12'
b(?
+ 7
5
+ 3
 1
1322
 14
+ 5
 1
1»2
+ 7
 1
V
b'
 1
(43)
(5P)
+ 7
 7
 14
 7
(421)
(3*1)
1
(32«)
(4P)
(321«)
(2*1)
(31^)
(2n»)
(2P)
(10
 7
+ 7
+ 21
+ 7
 7
 14
+ 7
 1
1
■
c. n.
54
426
A MEMOIR ON THE SYMMETRIC FUNCTIONS
[147
VIII (a). Runs on infrit.
(8)
(71)
(62)
(53)
(4')
(6P)
(521)
(431)
(42»)
(3''2)
(51')
(421«)
(3'1»)
(32n)
(2*)
(41')
(321')
(2'1»;
(31»;
(2'1«)
(21')
(P)
8
•
1
17
bh
26
eg
P6
35
df
125
be/
V5
4^
134
bde
2«4
1«24
b'^ce
P4 ; 23«
b*e \ cd*
1
1 1
1
: 1
1
•
—
1
1
1
1
1 i
1
1
— —
1 1 1
— .
+ 1 ...
+ 1
+ 4; ...
+ 1
+ 2
+ 6 , + 2
+ 1
+ 2
+ 5
+ 1214 5
+ 1
+ 4
+ 6
+ 12
+ 241+ 12
+ 1
« • •
• • •
• • •
+ 4
+ 17; ...
+ 1
+ 3
• • •
. 3
+ 7!+ 18
+ 46 + 12
+ 1
+ 3
+ 5
+ 6
+ 2
+ 11
+ 1«!+ 39
+ 84+31
+ 1
+ 1
+ 2
• • «
+ 16
+ 30
• • •
+ 10
+ 32
+ 20 1 + 55
+ 140 r 30
1
^ 4
+ 14
+ 53
+ 150
+ 114
+ 246 + 80
1
+ 1
+ 1
+ 8
+ 6
+ 28
+ 13
+ 15
+ 56
+ 51
+ 108
+ 20
+ 70
!+ 95
+ 315
+ 660 + 210
+ 1680 ' + 560
4
+ 56
+ lt)8
+ 336
+ 280
+ 420 1 + 840
■
1'3»
i
12«3
he'd
P23
iy'cd
1«3
bH
2*
c*
P2»
6V
P2«
6V
1«2
6«c
(8)
1
+
1
(71)
1 '

+
I
+
8
(62)
1
1 ;
+
+
1
+
6
+
28
(53)
+ 1
4
+
15
+
56
(4')
i+ 1
+ 2
+
6
+
20
+
70
(61»)
+ 1
1
• • •
+
2
+
13
+
56
(521)
+ 1
+ 5
• • •
+ 3
+
14
+
51
+
168
(♦31)
+
1
+ 3
+ 10
+ 4
+ 11
+
32
+
95
+
280
(42')
+ 1
+
2
+ 7
+ 20
+ 30
+ 6
+ 18
+
53
+
150
+
420
(3»2)
+ 2
+
5
+ 12
+ 12+31
+
80
+
210
+
560
(31')
• • •
+ 2
• • •
+ 3
+ 16
• • •
+ 6
+
30
+
108
+
336
(421')
+
5
+ 18
+ 55 + 12
+ 39
+
+
114
172
+
315
+
840
(3'1')
+ 4
+
12
+ 30
+ 80
+ 28
+ ()8
+
440
+
1120
(32'1)
+ 12
+
24
+ 58
+ 140
+ 48
+ 117
+
284
+
690
+
1680
(2«)
+ 28
+
48
+ 108
+ 240
+ 90
+ 204
+
468
+
1080
+
2620
(iV)
+ 6
+
12
+ 46
+ 140
+ 24
+ 84
+
246
+
660
+
1680
(321')
4 30
+
58
+ 141
+ 258
+ 340 + 108
+ 258
+
+
612
1008
+
1440
+
3360
(2'!')
+ 68
+
117
+ 570
+ 204
+ 453
+
2250
+
+
5040
6720
(31')
+ 80
+ 172
+
140
+ 340
+ 800
+ 240
+ 570
+
1320
+
3000
(2'1*)
+
284
+ 612
+ 1320
+ 468
+ 1008
+
2172
+
4680
+
10080
(21«)
+ 440
+
690
+ 1440
+ 3000
+ 1080
+ 2250
+
4680
+
9720
+
20160
(1»)
+ 1120
+
1680
+ 3360
+ 6720 ' + 2520
+ 5040
+
10080
+
20160 ^
40320
147]
OF THE ROOTS OF AN EQUATION.
427
VIII (6). Runs on iattk.
(8)
(71)
(62)
(53)
(4»)
(6P)
(521)
(431)
(42')
(3'2)
(51')
(421»)
(3n«)
(32*1)
(2')
(41^)
(321*)
(2»1»)
(31»)
(2'1*)
(21')
(1")
8
•
17
bh
26
P6
 b
+ 1
+ 2
+ 8
+ 4
 1
3
 9
 2
 5
+ 1
+ 4
+ 5
+ 5
• • •
 I
5
• • •
+ 1
35
df
+ 8
8
8
+ 7
4
+ 8
+ 1
+ 1
+ 8
7
3
9
+ 3
+ 6
 2
+ 3
3
+ 1
125
w
P5
^/
+ 8
 1
 2
3
 4
+ 1
4*
e«
+ 4
4
134
hde
2M
1«24
IH
23«
 8
+ 8
+ 8
 16
+ 9
16
 8
+ 21
88
+ 1' + 8
+ 2! + 2
+ .3 7
... ' + 4
 1  5
3+5
... 1 — 1
— 2
... + 1
1
1
+ 8
 1
 8
+ 9
+ 8
4
 10
+ 8
 8
+ 4
+ 4
4
4
+ 6
+ 4
+ 8
8
 4
+ 16
 6
+ 8
 8
 8
+ 1
+ 1
+ 8
 4
2
 4
• • •
 9
+ 4
 4
 4
+ 8
 8
+ 4
8
+ 1
+ 2
 3
 9
+ 4
 ]6
16
+ 9
+ 4
 8
+ 3
 10
+ 11
+ 9
+ 16
10
+ 4
+ 2
• • •
+ 10
 1
8
 8
+ 8
+ 8
 I
 4
4
• • •
+ 4
 2
+ 2
2
+ 1
 2
• • •
+ 2
+ 5
+ 4
 \
1
+ 8
_ 2
+ 3
 1
 4
■ • •
• • •
• • •
+ 1
+ 4
 4
+ 24
 10
 6
H 11
+ 4
 1
+ 1
+ 12
+ 24
 5
 9
 1
+ 2
 2
 17
• • •
 3
 4
+ 1
+ 1
1
1
1
1
1
1
1
•f 2
 8
 '2
+ 1
+ 2
+ 2
• • •
 3
32
+ 11
+ 8
+ 1
.
i16
+ 9
 4
+ 8
 1
 2
+ 20
 6
+ 1
1
 8
1
+ 1
^2
1'3«
6W
12«3
hi?d
1»23
1»3
h'd
+ 8
 1
2
• • •
• • •
+ 1
2*
c*
+ 2
 2
+ 2
2
+ 1
P2»
P2»
1«2
8
+ 1
(«)
+ 12
+ 24
32
16
+ 20
+ 1
(71)
 5
17
+ 11
+ 9
 6
(62)
 9
• • •
+ 8
4
+ 1
(53)
+ 3
+ 6
 3
+ 1
(61)
(521)
+ 2
+ 5
4
• • •
 5
1
+ 5
1
 1
 3
+ 1
(431), 2
+ 1
_ —
(42') ' + 1
i
•
1
1
!
1 1
!
I
54—2
428
A KEHOIR ON THE SYMMETRIC FUNCTIONS
[147
IX (a). Runs on to p. 430.
II
9
•
 1
18
6i
27
ch
1'7
 1
 2
 15
72
35
dg
 1
• • •
 5
 21
84
125
hcg
1»6
6V
45
135
hdf
2«5
1»25
1*5
14»
<9)
(81)
(72)
(63)
(64)
(71')
(621)
(531)
(4n)
1
I
(52')
(432)
(3')
*
 1
9
 1
 7
36
(61')
(521')
(431«)
(42'1)
(3'21)
(32')
 1
(51*)
1
• • •
(421»)
1
4
• • ■
(3n«)
 1
2
6
• • •
(32»1»)
 1
 2
5
 12
 2
(2*1)
 1
 4
 6
 12
 24
 9
(41)
 1
• • •
• • •
• • •
5
 21
• • ■
(321«)
 1
 3
• • •
 4
 9
 23
 58
 6
(2'1')
 3
 6
 3
 15
 24
 51
 108
 24
1
(31«)
 6
 19
• • •
 15
 30
 81
 204
 20
(2»1»)
 17
 36
 10
 50
 81
 172
 366
 70
(21')
 70
147
 35
161
252
 525
 1092
210
(!•)
252
504
 126
504
756
1512
3024
630
147]
OF THE ROOTS OF AN EQUATION.
429
II
(81
(72;
(63)
(54;
(71')
(62i;
(53i;
(4n;
(52«)
(432)
(3»)
(61»)
(521«)
(431«)
(42«i;
(3«2i;
(32'
(51*)
(42P)
(3*1»J
(32'1'^
(2*1
(4r
(321*)
(2»1»)
(31')
(2»1»)
(210
(!•)
234
cde
1»34
b^de
12H
be'e
1»24
b^ce
P4
b^e
3»
123»
6ccP
P3»
b^d"
2»3
c'd
1«2«3
•
1
1
2
1
• • •
2
1
3
3
8
1
3
6
6
15
1
• • •
• • •
• • •
• • «
• • •
1
5
• • •
• • •
2
• • •
5
1
3
10
• • •
2
6
7
19
1
2
7
20
• • •
5
17
 12
36
1
2
5
 12
30
3.
 13
30
 27
65
3
7
 12
 27
60
6
 27
64
 51
 120
■ • •
• • ■
• • •
4
21
• • •
• • •
6
• • •
12
• • •
3
7
 25
75
• • •
 12
42
 27
85
3
6
 17
 42
 110
6
 30
72
 64
 152
8
 19
 36
85
 200
 15
 65
 152
 120
 281
 22
 48
 78
 168
 360
 36
136
 300
 234
 516
• • •
 10
 20
 75
 225
• • ■
 30
 110
 60
 200
 22
54
 101
 241
 570
 36
 158
 372
 282
 656
 60
 129
 213
 459
 990
 93
 333
 720
 555
 1203
 60
 155
 270
 645
 1500
 90
 390
 920
 1740
 660
 1530
 166
 350
 565
 1200
 2550
 240
 820
1320
 2800
 455
 945
 1470
3780
3045
 6300
 630
2030
 4200
3150
 6510
1260
2520
7560
15120
 1680
5040
 10080
7560
 15120
4S0
A MfcMOIR ON THE SYMMKTRIC FUNCTIONS
[147
II
1*23
h'cd
P3 1
12*
6c*
1*2»
6V
1»2«
6V
r2
6»
(9)
1
(81)
1
9
(72)
1
7
36
(63)
1
5
21
84
(54)
1
3
10
35
126
(7P)
1
• • •
• • •
2
15
 72
(621)
1
6
• • •
3
17
70
252
(531)
4
6
15
4
15
50
161
504
(4n)
20
9
6
2i
70
210
 630
(52«)
9
30
24
' 81
252
756
(432)
22
60
90
22
60
! 165
455
 1260
(3«)
 36
36
93
 240
630
 1680
(6P)
3
19
• • •
6
36
147
504
(52 P)
23
81
12
51
 172
525
 1512
(43P)
54
 155
48
 129
 350
945
 2620
(42«1)
 101
 270
78
 213
i 565
 1470
 3780
(3«21)
 158
390
 136
 333
i 820
2030
 5040
(323)
 282
 660
 234
 555
 1320
 3150
 7560
(51*)
58
 204
 645
24
 108
1 366
 1092
 3024
(42P)
 241
 168
459
 1200
 3045
 7560
(3»1»)
 372
 920
 300
 720
 1740
 4200
 10080
(32n«)
 e.'ie
1530
516
 1203
' 2800
 6510
 15120
(2*1)
1140
 2520
 906
 2016
1 4500
 10080
 22680
(4P)
 570
 1500
 360
990
 2550
 6300
 15120
(321*)
1516
 2610
 3480
 1140
 2610
 5940
 13440
 30240
(2»P)
 5670
 2016
 4383
 9540
 20790
 45360
(3P)
 3480
 7800
 2520
 5670
1 12600
 27720
 60480
(2»P)
 5940
 12600
 4500
 9540
 20220
 42840
 42840
 90720
(2r)
 13440
 27720
 10080
 20790
 88200
 181440
(!•)
 30240
 60480
 22680
 45360
 90720
 181440
 362880
1 47]
OP THE ROOTS OF AN EQUATION.
431
IX ijb). Runs on to p. 432.
—
9
•
J
18
bi
27
ch
V7
35
d(/
125
beg
1'6
b'g
+ 9
 1
2
3
9
+ 1
+ 3
+ 4
+ 5
+ 2
+ 5
• • •
 1
4
5
5
• • •
• • •
+ 1
+ 5
• • •
• « •
• • •
 1
45
135
bdf
2«5
9
1«25
b'cf
1*5
hV
9
+ 1
+ 2
+ 3
+ 4
 1
3
4
• * •
2
• • •
• • •
+ 1
+ 4
• • •
• • •
• • •
• • •
 1
14«
b^
234
cde
1*34
(9)
 9
+ 9
+ 9
 9
+ 9
 18
+ 9
 9
 18
+ 27
 9
 18
+ 27'
(81)
+ 9
 1
 9
+ 1
 9
+ 10
+ 10
+ 9
 11
+ 5
+ 18
+ 4
 11
(72)
+ 91 9
+ 5
+ 29
+ 4
 9
+ 18
5
+ 9
 6
+ 9
20
(63)
+ 9
 9
 9
+ 9
+ 9
• • •
. 9
• • •
 9
+ 9
• • •
 9'
(54)
+ 9
 9
 9
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+ 9
 9
+ 18
+ 11
 2
 1
 7
11
 2
+ 13
+ 11
(71')
+ 1
+ 2
 1
+ 9
 3
+ 9
 10
2
+ 4
 5
 11
(621)
18 1+10
+ 4
 3
• • •
 8
+ 18
 10
 4
8
+ 11
 14
 4
+ 2
+ 13i
(531)
 18'+ 10
+ 18
 10
• • •
+ 9
 10
 2
 5
+ 15
+ 6
 Tj
(4«1)
 9
+ 5
+ 9
 5
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 11
+ 6
+ 1
• • •
+ 6
3
+ 2
+ 6
• • •
6
+ 3
• • •
2
+ 2
 1
+ 5
+ 2
 5
(52')
9+9
 5
 2
+ 9
 4
 1
 8
+ 2
+ 6
 6
+ 1
+ 6
• « •
+ 1
(432)
181+18
+ 4
 11
• • •
 4
 2
+ 2
 8
+ 3
+ 5
(3';
 3
+ 3
 1
+ 3
 2
 3
 6
+ 3
+ 3
+ 3
• • •
 3
• • •
(61')
+ 9
+ 1
 3
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+ 3
 9
+ 4
 4
+ 5
 5
(i521»)
+ 27
 11
 6
+ 4
+ 11
+ 5
+ 11
 7
+ 13
+ 3
+ 15
 15
 1
 5
+ 1
(m')
+ 27
 11
 20
+ 1
 13
 9
 y
+ 13
+ 12
 5
 2
 4
+ 2
+ 1
+ 3
 5
+ 1
+ 2
(+«1)
+ 27
 19
+ 1
+ 2
 11
1
(:V''21)
+ 27'
 19
 9
+ 12
+ 2
+ 18
. — 7
1
 7
• • •
+ 3
 1
(32»)
+ 9
+ 5
 3
+ 3
 2
+ 1
• • a
 1
(51')
 9
+ 1
+ 2
 1
 5
 3
+ 4
 4
+ 4
1
(421»)
36
 18
+ 12
+ 8
+ 12
 14
 4
+ 1
 1
(3n»)
+ 6
+ 11
 6
 3
+ 1
+ 4
 2
+ 2
 1
(32n»)
54
+ 30
+ 5
— 5
 9
 9
+ 3
+ 4
 1
 1
1
(2'1)
 9
+ 7
• • •
• • •
1
.
(41»)
+ 9
+ 45
 1
 2
+ 1
 3
+ 3
1
(321*)
 13
 10
+ 6
+ 3
 1
(2'1»)
+ 30
 14
+ 5
• • ■
 1
1
(31*)
 9!+ 1
+ 2
 1
 1
"
(2'1»)
 27
+ 7
1
(210
+ 9
 1
!
1 1
(!•)
 1
1
432
A MEMOIR ON THE SYMMETRIC FUNCTIONS
[147
__
12«4
bi^e
P24
b^ce
P4
b'^e
+ 9
 1
2
3
• • •
+ 1
+ 3
 1
3»
(P
3
+ 3
+ 3
6
+ 3
3
+ 3
+ 3
3
3
+ 3
 1
123»
bccP
P3»
b^(P
2»3
+ 9
9
+ 5
3
+ 1
+ 2
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+ 2
1
1«2>3
1*23
b*cd
1*3
b*d
9
+ I
+ 2
• • •
« • •
 1
12*
be*
9
+ 7
5
+ 3
 1
1«2»
6V
1»2»
V2
b'c
+ 9
1
I*
1
(9)
+ 27
36
+ 27
 18
54
+ 45
+ 30
27
(81)
 19
+ 12
19
+ 6
+ 30
13
14
+ 7
(72)
+ 1
+ 8
 13
+ 11
+ 5
10
+ 5
 1
(63)
 9
+ 12
+ 18
 3
 9
+ 3
 1
(54)
+ 3
 4
 7
 2
+ 4
• • •
(7P)
+ 5
 5
+ 12
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 9
+ 6
(621)
+ 12
 14
+ 1
 7
+ 1
+ 4
 1
(531)
 2
 4
+ 2
 1
(in)
+ 1
• • •
+ 3
• • •
(52«)
 6
+ 2
+ 3
 1
(432)
+ 2
• • a
 1
(3»)
• • •
• • •
(61')
 5
+ 5
(52P)
+ 3
 1
(43 1«)
 1
i
1
•
!
1
147]
OF THE ROOTS OF AN EQUATION.
433
X (a). Runs on to p. 436.
II
10
k
19
28
ci
P8
hH
37
dh
127
hch
1»7
6»A
46
^9
136
hdg
2»6
P26
h'cg
P6
h'g
5»
(10)
(91)
(82)
(73)
(64)
(5')
(8P)
(721)
+ 1
(631)
(541)
(62')
(532)
(4'2)
(43»)
(71')
(62 1»)
(53P)
•
(4n')
(52=1)
(4321)
(3'1)
(42')
(3»2')
(6P)
(52P)
(431')
,
(42n»)
(3»2P)
(32'1)
(2»)
+ 1
(5P)
+ 1
• • •
(421«)
+ 1
+ 2
+ 17
+ 90
+ 1
+ 4
• • •
(3'P)
f 1
+ 2
+ 6
• • •
(32'1»)
+ 1
+ 2
+ 5
+ 12
• • •
(2*1')
+ 1
+ 4
+ 6
+ 12
+ 24
+ 2
(41»)
+ 1
• • •
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+ 25
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(321»)
+ 1
+ 3
• • •
f 5
+ 11
+ 28
+ 70
• • •
(2«1')
+ 1
+ 3
+ 6
+ 4
+ 19
+ 30
+ 63
+ 132
+ 6
(310
• • •
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+ 22
• • •
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f 42
f 112
+ 280
• • •
(2'1')
+ 1
f 8
+ 45
+ 6
+ 20
+ 42
+ 15
+ 72
+ 115
+ 242
+ 510
+ 20
(21»)
+ 1
+ 10
+ 28
+ 92
+ 192
f 56
+ 252
+ 392
+ 812
+ 1680
+ 70
(1")
+ 120
f 360
f 720
+ 210
+ 840
+ 1260
+ 2520
+ 5040
+ 252 :
C. II.
55
434
A MEMOIB OK THE STMMETBIC FUNCTIONS
[147
II
145
235
cdf
1«35
12»5
hi?f
1»25
hhf
P5
24«
1»4»
6V
3>4
d^e
1234
hcde
(10)
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(541)
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(43«)
(71')
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(531')
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(52*1)
(4321)
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+ 1
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+ 1
+ 2
+ 2
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+ 1
• • •
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+ 1
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(3'21')
f 1
+ 2
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+ 12
+ 30
+ 2
+ 4
+ 5
+ 21
(32*1)
+ 1
+ 3
+ 7
+ 12
+ 27
+ 60
+ 7
+ 16
+ 12
+ 49
(2»)
+ 5
+ 10
+ 20
+ 30
+ 60
+ 120
+ 20
+ 45
+ 30
+ 110
(51")
• • •
• • •
• * •
...
+ 5
+ 26
• • •
• • •
• • •
• • •
(421')
• • •
• • •
f 4
+ 9
+ 32
+ 95
• • •
+ 6
• • •
+ 22
(3»1*)
• • •
+ 4
f 8
+ 22
+ 54
f 140
+ 6
+ 12
+ 12
+ 56
(32»1')
f 3
+ 11
f 26
f 48
+ 112
f 260
+ 480
+ 18
+ 42
+ 31
+ 128
(2n')
+ 14
f 32
+ 68
+ 108
+ 228
+ 53
+ 114
+ 80
+ 284
(41«)
• • •
• • •
+ 15
+ 30
+ 111
+ 330
• • •
+ 20
• • •
+ 60
(321»)
+ 10
+ 35
4 85
+ 156
+ 368
+ 860
+ 50
+ 120
+ 80
+ 335
(2'1<)
+ 42
+ 99
+ 210
+ 339
+ 720
+ 1530
+ 144
+ 306
+ 213
+ 735
(310
+ 35
+ 105
f 266
+ 462
+ 1092
+ 2620
+ 140
+ 360
+ 210
+ 876
(2»1«)
+ 130
+ 296
+ 622
+ 990
+ 2082
+ 4380
+ 400
+ 840
+ 570
+ 1900
(21')
+ 406
+ 868
f 1792
+ 2772
+ 5712
+ 11760
+ 1120
+ 2310
+ 1540
+ 4900
(1")
+ 1260
+ 2520
+ 5040
+ 7560
+ 15120
+ 30240
+ 3150
+ 6300
+ 4200
+ 12600
147]
OF THE ROOTS OF AN EQUATION,
435
P34
2»4
1«254
6Vc
1*24
1«4
13'
223»
1>23«
1*3»
h'd^
12»3
h&d
•
f 1
+ 1
• • •
+ 1
+ 4
+ 3
+ 1
+ 2
+ 6
+ 7
+ 1
f 2
+ 5
+ 12
+ 12
+ 1
• • •
• • •
• • •
• • •
+ 1
+ 6
• • •
• • •
+ 2
• • •
+ 1
+ 4
+ 15
• • •
+ 2
+ 8
+ 7
+ 1
f 2
f 6
+ 20
+ 2
+ 4
+ 12
+ 16
+ 1
■ • •
f 2
+ 9
+ 30
• • •
+ 5
+ 22
f 12
+ 3
+ 3
+ 8
+ 22
+ 60
+ 3
+ 8
+ 21
+ 56
+ 49
f 87
+ 6
+ 6
+ 15
+ 36
+ 90
f 10
+ 18
+ 42
+ 96
+ 10
+ 6
+ 18
+ 48
+ 120
f 6
+ 15
+ 42
+ 115
+ 87
+ 18
+ 15
+ 34
+ 78
+ 180
+ 18
+ 34
+ 80
+ 188
+ 156
• • •
• • •
• • •
+ 4
f 25
• • •
• • •
• • •
+ 6
• • •
+ 3
• • •
+ 7
+ 32
+ 76
+ 111
• • •
• • •
+ 12
+ 54
+ 27
+ 9
+ 10
+ 27
+ 215
+ 6
f 18
+ 48
+ 132
+ 112
+ 27
+ 18
+ 54
+ 149
f 390
+ 15
+ 34
f 99
+ 270
f 198
+ 48
+ 42
+ 99
+ 236
+ 570
f 42
+ 80
+ 186
+ 436
+ 358
+ 112
+ 87
+ 198
+ 450
+ 1020
+ 87
f 156
+ 358
+ 820
+ 645
+ 240
+ 180
+ 390
+ 840
+ 1800
+ 180
+ 310
+ 680
+ 1500
+ 1170
+ 10
• • •
+ 20
+ 95
f 330
• • •
• • •
f 30
+ 140
+ 60
+ 76
+ 48
f 149
+ 416
+ 1095
+ 36
+ 78
+ 236
f 650
+ 450
+ 132
+ 115
+ 270
+ 650
+ 1580
+ 96
f 188
+ 436
+ 1032
+ 820
+ 294
+ 228
+ 523
+ 1196
+ 2730
+ 210
+ 370
+ 844
+ 1920
+ 1479
+ 612
+ 468
+ 1008
+ 2172
+ 4680
+ 444
+ 740
+ 1604
+ 3480
+ 2688
+ 215
+ 120
+ 390
+ 1095
+ 2850
+ 90
+ 180
+ 570
+ 1580
+ 1020
+ 775
+ 585
+ 1340
+ 3050
+ 6900
+ 510
+ 880
+ 2000
+ 4520
+ 3390
+ 1566
+ 1194
+ 2547
+ 5436
+ 11610
+ 1092
+ 1776
h 3792
+ 8100
+ 6180
+ 2030
+ 1470
+ 3360
+ 7560
f 16800
+ 1260
+ 2100
+ 4760
+ 10640
+ 7770
+ 3990
+ 3015
+ 6330
f 13290
+ 27900
f 2700
+ 4280
+ 8980
f 18840
+ 14220
+ 10080
+ 7560
+ 15540
+ 31920
+ 65520
+ 6720
+ 10360
+ 21280
+ 43680
+ 32760
+ 25200
+ 18900
+ 37800
+ 75600 + 151200
4 16800
f 25200
f 50400
f 100800
+ 75600
552
436
A MEMOIR ON THE SYMMETRIC FUNCTIONS
[147
II
i
l»2«3
9c'd
P23
b^cd
V3
2»
2»3>
b^c*
P2»
6V
1«2»
6V
1»2
110
6"
(10)
4
1
(91)
4
1
4
10
(82)
4
1
4
8
4
45
(73)
•
4
1
4
6
4
28
4
120
(64)
4
1
4
4
4
15
4
56
4
210
(5')
4
1
4
2
4
6
4
20
+
70
4
252
(81')
4
4
1
t
• • •
• • •
• • •
4
2
4
17
4
90
(721)
+
1
• • •
• • •
4
3
4
20
4
92
4
360
(631)
+
1
+
5
4
21
• • •
4
4
4
19
4
72
+
252
4
840
(541)
+
3
+
10
4
35
4
5
4
14
4
42
4
130
4
406
4
1260
(62»)
+
2
■+
11
4
42
• • •
4
6
4
30
4
115
4
392
4
1260
(532)
+
11
+
35
4
105
4
10
4
32
4
99
4
296
4
868
4
2520
(4*2)
+
18
+
50
4
140
4
20
4
53
4
144
4
400
4
1120
4
3150
(43')
+
31
+
80
4
210
4
30
4
80
4
213
4
570
4
1540
+
4200
(7P)
• • •
+
3
4
4
22
112
• • •
• • •
4
6
4
42
4
192
4
720
(62 1»)
+
5
+
28
• • •
4
12
4
63
4
242
4
812
4
2520
(sai*)
+
26
+
85
4
266
4
20
4
68
4
210
4
622
+
1792
4
5040
(4'1»)
+
42
+
120
4
350
4
45
4
114
4
306
4
840
4
2310
4
6300
(52n)
+
48
+
156
4
462
4
30
4
108
4
339
4
990
4'
2772
4
7560
(4321)
+
128
+
335
4
875
4
110
4
284
4
735
4
1900
4
4900
4
12600
(3'1)
+
210
+
510
4
1260
4
180
+
444
4
1092
4
2700
4
6720
4
16800
(42')
+
228
+
585
4
1470
4
180
4
468
4
1194
4
3015
4
7560
4
18900
(3»2'')
+
370
+
880
4
2100
4
310
4
740
4
1776
4
4280
4
10360
4
25200
(61')
+
12
+
70
4
280
• • •
4
24
4
132
4
510
4
1680
4
5040
(521»)
+
112
4
368
4
1092
4
60
4
228
4
720
4
2082
4
5712
4
15120
(43P)
+
294
+
775
4
2030
4
240
4
612
4
1566
4
3990
4
10080
4
25200
(42n»)
+
523
+
1340
4
3360
4
390
4
4
1008
4
2547
4
6330
4
15540
4
37800
(3»21')
+
844
+
2000
4
4760
4
680
1604
4
3792
4
8980
4
21280
4
50400
(324)
+
1479
+
3390
4
7770
4
1170
4
2688
4
6180
4
14220
4
32760
4
75600
(2')
+
2580
+
5700
4
12600
4
2040
4
4530
4
10080
4
22500
4
50400
4
113400
(51')
+
260
4
860
4
2520
4
120
4
480
4
1530
4
4380
4
11760
4
30240
(121*)
+
1196
+
3050
4
7560
4
840
1500
+
2172
4
5436
4
13290
4
31920
4
75600
(3»1*)
4
1920
4
4520
4
10640
4
4
3480
4
8100
4
18840
4
43680
4
100800
(32n>)
+
3358
4
4
7610
12720
4
4
17220 4
2580
; 4
5844
4
13212
4
29820
4
67200
4
151200
(2n»)
+
5844
27720
4
4530
4
9876
4
21564
4
47160
4
103320
4
226800
(41«)
+
2730
4
6900
16800
4
1800
4
4680
+
11610
4
27900
4
65520
4
151200
(321*)
+
7610
4
4
17000
28260
4
37800 1 4
5700
+
12720
4
28260
4
62520
4
137760
4
302400
(2»1*)
f
13212
4
60480 4
10080
4
21564
+
46152
4
98820
4
211680
4
453600
(31')
+
17220
4
37800
4
82320
4
12600
4
27720
4
60480
4
131040
4
282240
4
604800
(2n»)
+
29820
4
62520
4
4
4
131040
+
22500
4
47160
4
98820
4
207000
+
433440
4
907200
(21')
67200
4
137760
302400
282240
4
50400
4
103320
4
211680
4
433440
4
887040
4
1814400
(V)
+
151200
4
604800
4
113400
4
226800
4
453600
4
907200
4
1814400 1 +
362880C
ii
147]
OF THE BOOTS OF AN EQUATION.
437
10
k
19
28
ci
P8
bH
£(6).
37
dh
Run
127
bch
20
s on 1
P7
ko p. 
46
139.
136
bdg
2>6
P26
b^'cg
4 30
 12
1*6
b'g
5»
145
be/
(10)
10
+ 10
+ 10
 10
+ 10
4 10
4 10
20
 10
10
+
5
20
(91)
+ 10
 1
10
4 6
4. 1
 10
4 11
 1
 2
 10
4 11
4 10
4 1
4. 2
—
5
+ 11
(82)
+ 10
 10
4 2
 10
4 4
10
4 20
 6
 6
—
5
+ 20
(73)
+ 10
 10
 10
4 10
4 11
 1
 3
 10
 1
4 10
 9
4 3
—
5
+ 20
 4
(64)
+ 10
 10
 5
 10
4 10
 10
4 20
 10
4 14
 4
 2
 6
4 4
4
5
10
(5*)
+ 5
 5
4 5
 5
4 10
5
5
4 10
4 5
 15
+ 5
 15
(8P)
10
+ 1
^ ^
 1
4 10
 3
4 1
4 10
 11
 2
 4
+ 4
 1
4
5
 11
31
7
(721)
 20
+ 11
4 4
 3
 1
 8
4 3
+ 20
 10
4 11
 3
 4
4
10
(631)
 20
+ 11
+ 11
4 20
+ 20
 11
 1
10
+ 4
4
 4
 8
4 15
4
10
15
(541)
 20
 11
 2
4 20
31
+ 11
 4
 7
 8
4 18
4 6
 5
4 23
(62*)
 10
+ 10
 6
+ 10
 4
4 2
 2
 8
• • ■
 2
 5
+
5
 8
(532)
 20
4 20
4 4
4 2
 12
1
 3
4 5
4 20
 19
 4
4. 10
4. 15
—
15
+ 10
(4«2)
 10
+ 10
 6
4 10
 12
+ 6
4 3
14
4 4
 6
• • •
+
5
4 4
(43«)
10
+ 10
 1
+ 10
 10
 11
4 1
 2
4. 13
 4
 3
• • •
4
5
 8
(7P)
+ 10
2
6
4 1
4 4
 3
4. 3
 1
 10
4. 4
4 2
 4
4. 1
—
5
4. 11
(62P)
+ 30
 12
9
+ 11
 4
 6
4 15
4. 6
15
4 4
—
15
4 18
(53P)
+ 30
 12
22
 11
4 2
4. 12
 9
4. 13
5
 6
4 15
4 10
 19
4 5
4
10
 12
(4«P)
+ 15
 6
4 6
 15
4 17
 6
4 9
 3
 1
+ 1
• • •
4
5
 8
(52«1)
+ 30
21
4 5
 9
4 12
 5
 18
4 18
+ 4
17
4 5
4
10
 1
(4321)
+ 60
42
28
4 26
4. 3
4 21
 12
4 12
 15
 8
4 7
• • •
—
5
• • •
(3'1)
+ 10
7
 10
4 7
+ 11
 4
• • •
4 2
 7
• • •
4 4
• • •
• • •
 5
4 5
(42»)
+ 10
 10
4 6
4 2
 10
4 4
 2
+ 10
 4
4 2
• • •
—
5
• • •
(3«22)
4 15
 15
+ 1
+ 13
+ 13
4 1
4 7
4 6
 7
• • •
 9
4 3
4 2
• • •
• • •
4
5
 1
(6P)
 10
4 2
+ 8
 1
+ 3
 3
+ 1
4 4
 4
 2
4 4
 1
4
5
 5
(521»)
 40
 5
+ 12
 14
 16
 23
+ 5
4. 16
19
 8
4. 19
 5
—
5
4 1
(43P)
 40
+ 24
 13
 9
21
4 12
+ 6
 8
4 5
• • •
 1
—
5
4 5
(42212)
 60
4 33
+ 33
4 4
4 18
+ 9
 12
4 3
+ 8
4
4
5
 1
(322P)
 60
4 28
24
4. 9
• • •
• • •
4 6
 4
• • •
4
5
 3
(32n)
 40
+ 31
 8
 2
 7
 2
4 5
• • •
4. 8
 3
• • •
■ • •
—
5
4 1
(2»)
 2
+ 2
• • •
4. 2
• • •
• • •
 2
• • •
• • •
• • •
4
1
(5P)
+ 10
 1
2
+ 1
 3
+ 3
 1
 4
4. 4
4 2
 4
4 1
(42P)
+ 50
 14
 10
+ 6
15
4 17
 6
4 4
1
2
4 1
(3«P)
+ 25
 7
 13
4. 7
4 3
4 12
 1
• • •
4 2
 2
4 1
(32n»)
+ 100
46
 12
4. 14
 5
• • •
 4
4. 1
(2n')
+ 25
 16
+ 9
4 2
• • ■
 4
+ 3
• • •
• • •
4 1
(4P)
 10
+ 1
 1
 3
+ 1
(32P)
 60
4 15
+ 12
 7
 3
4 1
(2'P)
 50
+ 20
6
• • •
4. 1
(3F)
+ 10
1
 2
+ 1
(2n«)
+ 35
 8
4 1
(2P)
 10
4. 1
1
1
(PO)
+ 1
438
A MEMOIR ON THE SYMMETRIC FUXCTIONS
[147
—
235
cdf
1>35
hHf
+ 30
12
12«5
6cy
P25
1»5
24»
C6*
1H»
6V
3«4
1234
bcde
+ 60
42
P34
b^de
2»4
c'e
1»2H
1*24
b^ce
b^e
(10)
20
+ 20
+ 30
40
+ 10
 10
+ 15
 10
+ 10
40
+ 10
60
+ 50
10
(91)
21
+ 13
 1
+ 10
+ 2
 6
+ 13
10
+ 33
14
+ 1
(82)
+ 4
22
+ 2
+ 8
 2
 11
+ 10
11
 2
+ 5
28
+ 24
+ 12
+ 6
+ 4
 10
+ 2
(73)
 1
 9
 9
+ 12
 3
+ 10
15
+ 3
+ 12
 10
+ 18
 15
+ 3
(64)
+ 20
 6
 18
+ 16
4
 14
+ 5
+ 9
+ 5
 8
+ 10
 5
 12
+ 4
• • •
• • •
 1
 3
(5^)
 15
+ 10
+ 10
 5
• • •
 5
 5
+ 5
• • •
(8P)
 12
+ 12
+ 5
 5
+ 1
+ 3
 6
+ 6
 10
+ 26
13^
+ 2
 9
+ 6
(721)
 3
+ 13
+ 12
 14
 12
+ 17
+ 1
+ 21
 16
+ 4
23
+ 17
(631)
 19
+ 15
+ 18
 19
+ 4
+ 4
+ 4
 3
+ 13
 15
+ 5
• • •
• • •
+ 3
 1
(541)
+ 10
 12
 1
+ 1
• • •
 8
8
• • •
+ 5
 1
• ■ •
(62»)
 4
+ 10
 4
+ 2
+ 4
 8
+ 2
+ 10
 1
 4
+ 1
; 2
 8
+ 5
+ 4
 3
• • •
 4
+ 8
 2
(532)
+ 17
 13
+ 5
• • •
 12
+ 1
3
+ 3
 6
 1
• • •
+ 4
 2
■ • •
(4«2)
 12
+ 2
• • •
• • •
+ 5
+ 19
• • •
• • •
+ 2
+ 2
+ 6
 6
+ 2
 2
• • •
• • •
(43«)
+ 1
 3
+ 3
 1
• • •
• • •
• • •
• ■ •
+ 1
(7P)
+ 5
 5
 5
 17
 1
 4
+ 3
12
+ 7
+ 6
 2
+ 9
 6
(621«)
+ 15
 4
 19
+ 1
 3
 3
 1
+ 2
 2
 4
+ 1
(53 1«)
+ 3
+ 2
1
+ 2
+ 4
 2
+ 1
(4n«)
f 1
+ 2
 1
• • •
 3
+ 3
+ 3
 3
• • •
+ 1
(52n)
 13
+ 2
+ 9
 3
 3
• • •
• • •
• • •
• • •
 5
+ 1
+ 2
 1
 3
+ 3
 3
+ 1
(4321)
+ 5
+ 4
+ 4
 3
+ 1
+ 1
(3'1)
 1
• • •
• • •
 2
 2
+ 1
• • •
(42')
+ 4
 2
■ • •
+ 1
(3>2*)
 2
 5
+ 5
 1
 2
+ 1
• • •
• • •
(61*)
+ 5
+ 5
+ 1
(52 P)
1
 3
(43P)
2
+ 1
(42n')
+ 1
(3»21>)
147]
OF THE ROOTS OF AN EQUATION.
439
13»
hd^
2'3«
c^d^
1«23«
h^cd^
1*3«
h*d^
12»3
P2«3
P23
h^cd
r3
Vd
2»
c»
2
+ 2
2
+ 2
2
+ 1
2>3»
+'25
1*2»
6V
1«2»
6V
P2
110
+ 1
+ 10
+ 15
60
+ 33
+ 25
40
+ 100
60
+ 10
50
+ 35
 10
 7
 15
 7
+ 31
 46
+ 15
 1
 16
+ 20
 8
+ 1
 10
+ 1
+ 28
 13
 8
 12
+ 12
 2
+ 9
 6
+ 1
h 11
+ 6
 24
+ 3
 2
+ 12
 3
• • •
 4
+ 1
+ 2
 9
• • •
+ 2
+ 8
 4
• • •
• • •
+ 1
 5
+ 5
+ 5
• • •
 5
• ■ •
• • •
• • •
+ 7
+ 7
 21
+ 7
 7
+ 14
 7
+ 1
 4
 7
+ 9
 1
+ 5
 5
+ 1
 7
+ 3
+ 6
 2
 3
+ 1
+ 5
 1
 3
• • •
+ 1
+ 4
+ 2
 4
+ 1
 1
 2
+ 1
 2
+ 1
+ 1
440
[148
148.
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO
EQUATIONS.
[From the Philosophical Transactions of the Royal Society of London, vol cxLvn. for the
year 1857, pp. 703—715. Received December 18, 1856,— Read January 8, 1857.]
The Resultant of two equations such as
(a, 6, ...50?, y)"* = 0,
(p, q, ...5a?, y)'*=0,
is, it is well known, a function homogeneous in regard to the coefficients of each
equation separately, viz. of the degree n in regard to the coefficients (a, 6, ...) of
the first equation, and of the degree m in regard to the coefficients (jd, g, ••) of
the second equation; and it is natural to develope the resultant in the form
AilP + A'^.'P' + &c., where A, A\ &c. are the combinations (powers and products) of
the degree n in the coefficients (a, 6, ...), P, P', &c. are the combinations of the
degree m in the coefficients (p, g, ...), and k, Id, &c. are mere numerical coefficients.
The object of the present memoir is to show how this may be conveniently eflfected,
either by the method of symmetric functions, or from the known expression of the
Resultant in the form of a determinant, and to exhibit the developed expressions for
the resultant of two equations, the degrees of which do not exceed 4. With respect
to the first method, the formula in its best form, or nearly so, is given in the
Algebra of Meyer Hirsch, [for proper title see p. 417], and the application of it is very
easy when the necessary tables are calculated: as to this, see my "Memoir on the
Symmetric Functions of the Roots of an Equation "(^). But when the expression for the
Resultant of two equations is to be calculated without the assistance of such tables,
it is I think by far the most simple process to develope the determinant according
to the second of the two methods.
» Philosophical Transactiont, 1857, pp. 489—497, [147J.
148] MEMOIR ON THE RESULTANT OF A SYSTEM OP TWO EQUATIONS. 441
Consider first the method of symmetric ft^^ctioQS, and to fix the ideas, let the
two equations be
(a, b, c, cH^x, yY = 0,
(jp, q, r "^x, yf = 0.
Then writing
so that
(a, 6, c, d\l, zY = a(l ^ az)(l ^ fie)(l yzl
A = a + fi + y =(1),
 f = <^7  (I').
the Resultant is
ip. 3. rSa, 1)' • ip, q, »*M 1)' • (p. q. rJi% lY,
which is equal to
r' + qr' (a + /5 + f) + pr' (a* + ^ + i*) + pqr (a?/5 + afi* + 0^ + fiy' + '^ + i'a) + Sk.;
or adopting the notation for symmetric functions used in the memoir above referred
to, this is
{ r»
{ + qr' (1)
f+pr^ (2)
X+q'r (P)
j+pqr (21)
1+9* (1*)
'+ph (2»)
+pq' (2P)
{+p'q (2*1)
l+P' (2*) .
{
the law of which is best seen by dividing by r» and then writing
? = [!]. f = [2].
and similarly,
^ = [1']. f=[21].&c;
56
c. ir.
442
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
[148
the expression would then become
1 + [1] (1) + [2] (2) + [V] (V) + [21] (21) + [V] (P) + [2»] (2«) + [21»] (21«) + [2»1] (2n) + [2»] (2'),
where the terms vdthin the [ ] and ( ) are simply all the partitions of the numbers
1, 2, 3, 4, 5, 6, the greatest part being 2, and the greatest number of parts being 3.
And in like manner in the general case we have all the partitions of the numbers
1, 2, 3, ...mn, the greatest part being n, and the greatest number of parts being tn.
The symmetric functions (1), (2), (1*), &c. are given in the Tables (6) of the
Memoir on Symmetric Functions, but it is necessary to remark that in the Tables
the first coefficient a is put equal to unity, and consequently that there is a power
of the coefficient a to be restored as a factor: this is at once eflfected by the con
dition of homogeneity. And it is not by any means necessary to write down (as for
clearness of explanation has been done) the preceding expression for the Resultant;
any portion of it may be taken out directly from one of the Tables (6). For instance,
the bracketed portion
+ pqr (211
+ ?' (1').
which corresponds to the partitions of the number 3, is to be taken out of the
Table 111(6). as follows: a portion of this Table (consisting as it happens of consecutive
lines and columns^ but this is not in general the case) is
be
+ 8
1
1
= d
(ii)
if in this we omit thd sigh =, and in the outside line write for homogeneity ad
instead of d, and in the outside column, first substituting q, p for 1, 2, then write
for homogeneity pqr instead of pq, we have
ad he
pqr
+ 3
1
1
1
viz. pjr X (+3ad — 16c) + g'(— lad), for the value of the portion in queistion; this is
equivalent to
vqr q^ ^^'''^^'^
, or as it may be mdre convenielitiy Writteti,
be
in which form it constitutes a part of the expression given in the sequel for the
Resultant of the two functions in question; and similarly the remainder of the expres
sion is at once derived from the Tables (b) I. to VI.
+ 3 1
1
148]
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
443
As a specimen of a mode of verification, it may be remarked that the Resultant
qui invariant ought, when operated upon by the sum of the two operations,
3a36 + 263o + c9rf and 2pdq + qdr,
to give a result zero. The results of the two operations are originally obtained in the
forms in the first and second columns, and the first column, and the second column,
with all the signs reversed, are respectively equal to the third column, and conse
quently the sum of the first and second columns vanishes, as it ought to do.
\:
^d.
\
o*.
\
Next to explain the second method, viz. the calculation of the resultant from the
expression in the form of a determinant.
Taking the same example as before, the resultant is
a, 6, c, d
a, 6, c, dy
56—2
444
MEMOIR ON THE BBSXTLTANT OF A SYSTEM OF TWO EQUATIONS.
[148
which may be developed in the form
+ 12 . 345 }
 13 . 246 j
+ 14 . 235
+ 23 . 145
 15 . 234
 24 . 135
where 12, 13, &c.* are the terms of
(
and 123, &c. are the terms of
(
+ 25 . 134
+ 34 . 126
36 . 124}
445 . 123}
a, 6, c, d )
a, b, c, d
p, q. r
viz. 12 is the determinant formed with the first and second oolumns of the upper
matrix, 123 is the determinant formed vdth the first, second and third columns of
the lower matrix, and in like manner for the analogous symbols. These determinants
must be first calculated, and the remainder of the calculation may then be arranged
as follows: —
\:
c^.
\
\
r X <
J
> ^ <
\
«v.
«♦
148]
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
445
where it is to be observed that the figures in the squares of the third column are
obtained from those in the corresponding squares of the first and second columns by
the ordinary rule for the multiplication of determinants, — ^taking care to multiply the
dexter lines (ie. lines in the direction \) of the first square by the sinister lines
(i.e. lines in the direction/) of the second square in order to obtain the sinister lines
of the third square. Thus, for instance, the figures in the square
are obtained as follows, viz. the first sinister line (h 3, — 1) by
(1, +l)(2, +1)= 2 + l = + 3,
(1, +1)(+1, 0) = l + = l,
and the second sinister line (— 1, 0) by
(0, l)(2, +1) = 0.1=1,
(0, l)(^l, o) = o + o= 0.
I have calculated the determinants required for the calculation, by the preceding
process, of the Resultant of two quartic equations, and have indeed used them for
the verification of the expression as found by the method of symmetric functions; as
the determinants in question are useful for other purposes, I think the vahies are
worth preserving.
Table of the Determinants of the Matrices,
and
(
(
a,
6,
c,
d, e )
a,
6,
c,
d,
e
a,
6,
c,
d,
e
a,
6,
c,
d,
«,
f
P»
tf»
r,
8, t )
P>
?»
r.
s,
t
P*
q>
r,
«.
t
P>
9»
r,
s,
t
HEHOIB OH THE RESULTANT OF A SYSTEH OF TWO EQUATIONS.
[148
arranged in the form adapted for the calculation of the Resultant of the two qiuutic
equations (a, b, c, d, e^x, yf = 0, and (p, q, r, s, t'Sx, y)* = 0, viz.
148] MEMOIR ON THE BE8ULTANT OF A SYSTEM OF TWO EQUATIONS. 447
448 MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. [l48
/^■^
■r %
/\
148] MEMOIR ON THE RESHLTANT OF A SV8TGM OF TWO EQUATIONS. 449
The Tables of the Resultants of two uquatious which I have calculated are as
450
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
[148
Table (2, 2).
Resultant of
(a, 6, c\x, yY,
Table (3, 2).
Resultant of
(a, 6, c d^x, yy,
(/>, q. r "^x, yy.
Table (4, 2).
Resultant of
(a, 6, c, d, e^x, yy,
(p, 9, r^x, yy.
\
\/
148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
Table (*. 3).
Resultant of
(a, 6, c, d. e$x, y)*,
ip, q, r, sXx, yf.
Table (3, 3).
Resultant of
(a, b. c. d5», sr.
(^5, r, .5., y).
yA
'0
'>\<>
MEMOIR ON THE BE8ULTANT OF A SYSTEM OF TWO EQUATIONS.
Table (4, 4),
RfsiiUiint of
(a, 6, c, d, e^x; yY.
(p. q, r. s, l\x. yy.
[148
148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 453
454
[149
149.
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS OF CERTAIN
SYSTEMS OF TWO EQUATIONS.
[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the
year 1857, pp. 717 — 726. Received December 18, 1856, — Read January 8, 1857.]
Suppose in general that ^ = 0, ^ = 0, &c. denote a system of (n — 1) equations
between the n variables {x, y, z, ...), where the functions <f>, y^,' &c. are quantics ^e.
rational and integral homogeneous functions) of the variable& Any values {xi, yi, Zi,...)
satisf}dng the equations, are said to constitute a set of roots of the system; the roots
of the same set are, it is clear, only determinate to a common factor pris, Le. only
the ratios inter se and not the absolute magnitudes of the roots of a set are deter
minate. The number of sets, or the degree of the system, is equal to the product
of the degrees of the component equations. Imagine a function of the roots which
remains unaltered when any two sets (o^i, yi, Zi, ...) and (^, y,, z^, ...) are interchanged
(that is, when Xi and x^, yi and y,, &c. are simultaneously interchanged), and which is
besides homogeneous of the same degree as regards each entire set of roots, although
not of necessity homogeneous as regards the different roots of the same set; thus,
for example, if the sets are (xi, yi), (x^, y,), then the functions XyX^, XtJ/^ + x^i, y^y^
are each of them of the form in question; but the first and third of these functions,
although homogeneous of the first degree in regard to each entire set, are not homo
geneous as regards the two variables of each set. A function of the abovementioned
form may, for shortness, be termed a symmetric function of the roots; such function
(disregarding an arbitrary factor depending on the common factors which enter implicitly
into the different sets of roots) will be a rational and integral function of the coefficients
of the equations, i.e. any symmetric function of the roots may be considered as a
rational and integral function of the coefficients. The general process for the investi
gation of such expression for a symmetric function of the roots is indicated in Pro
fessor Schlafli's Memoir, "Ueber die Resultante eines Systemes mehrerer algebraischer
149] ON THE SYMMETRIC FUNCTIONS OF THE ROOTS, &C. 455
Gleichungen," Vienna Transactions, t. iv. (1852). The process is as follows: — Suppose
that we know the resultant of a system of equations, one or more of them being
linear; then if ^ = be the linear equation or one of the linear equations of the
system, the resultant will be of the form 0i^..., where <^i, ^, &c. are what the
function <}> becomes upon substituting therein the different sets (a?i, yi, 2^1 ...)> (^i ya* ^^a**)
of the remaining (n — 1) equations ^ = 0, X ~ ^' ^* > comparing such expression with
the given value of the resultant, we have expressed in terms of the coefficients of the
functions y^, x* ^j certain symmetric functions which may be caHed the fundamental
symmetric functions of the roots of the system ^ = 0, % = 0, &c. ; these are in fact
the symmetric functions of the first degree in respect to each set of roots. By the
aid of these fundamental symmetric functions, the other symmetric functions of the
roots of the system ^ = 0, x ~ ^» ^ ^^7 ^ expressed in terms of the coefficients,
and then combining with these equations a nonlinear equation * = 0, the resultant
of the system 4> = 0, ^^ = 0, x^^> ^ ^^^ ^ what the function *i*i . . . becomes, upon
substituting therein for the different symmetric functions of the roots of the system
•^ = 0, x = 0, &c. the expressions for these functions in terms of the coefficients. We
thus pass from the resultant of a system ^ = 0, '^ = 0, x = 0, &c., to that of a system
4> = 0, V^ = 0, X " ^> ^» ^^ which the linear function <f> is replaced by the nonlinear
function *. By what haa preceded, the symmetric functions of the roots of a system
of (n — 1) equations depend on the resultant of the system obtained by combining the
(n—l) equations with an arbitrary linear equation ; and moreover, the resultant of any
system of ri equations depends ultimately upon the resultant of a system of the same
number of equations, all except one being linear; but in this case the linear equations
determine the ratios of the variables or (disregarding a common factor) the values of
the variables, and by substituting these values in the remaining equation we have the
resultant of the system. The process leads, therefore, to the expressions for the
synmietric functions of the roots of any system of (w— 1) equations, and also to the
expression for the resultant of any system of n equations. Professor Schlafli discusses
in the general case the problem of showing how the expressions for the fundamental
symmetric functions lead to those of the other symmetric functions, but it is not
necessary to speak further of this portion of his investigations. The object of the
present Memoir is to apply the process to two particular cases, viz. I propose to
obtain thereby the expressions for the simplest symmetric functions (after the funda
mental ones) of the following systems of two ternary equations; that is, first, a linear
equation and a quadric equation ; and secondly, a linear equation and a cubic
equation.
First, consider the two equations
(«, 6, c, /, g, h^x, y, zy = 0,
(a, /8, yj^x, y, z) = 0,
and join to these the arbitrary linear equation
(f , V> ?$^, y, ^) = 0,
456
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS
[149
then the two linear equations give
and substituting in the quadratic equation, we have for the resultant of the three
equations,
(a, 6, c,/ g, hJi^^^ryr,, 7?  «?, a^/9f)^ = 0,
which may be represtented by
(a, b, c, f, g, h$f 17, f)> = 0,
where the coefficients are given by means of the Table.
a
b
c
f
2)8y
9,
h
a=>
+ >*
+ fi'
1 (a
b =
+ y'
+ a'
2ya
(J*)
c =
+ /3»
+ o»
a*
+ afi
(D
f =
Py
2(i»0
g =
ya
+ a)8
P"
+ ;3y
2«0
h=
aP
+ ya
^Py
y"
2(^)
viz. a = 67* + C)8*  2/597, &c.
But if the roots of the given system are
then the resultant of the three equations will be
and comparing the two expressions, we have
a =
'X1X2
i
b =
yiy«
»
c =
2^18^2
i
2f =
yi^2
+ y^i>
2g =
ZlX^
^z^i.
2h =
^iVi
+ arjyi,
which are the expressions for the six fundamental symmetric functions, or symmetric
functions of the first degree in each set, of the roots of the given system.
By forming the powers and products of the second order a', ab, &c., we obtain
linear relations between the symmetric fimctions of the second degree in respect to
each set of roots. The number of equations is precisely equal to that of the
149]
OF CERTAIN SYSTEMS OF TWO EQUATIONS.
457
symmetric functions of the form in question, and the solution of the linear equations
gives —
a* = x^x^,
b* = yi V.
he = ViZ^^t ,
ca = ZiXiZ^ ,
ab = x^iXttf,,
4f»
4g»
4h»
2bc =
2ca =
2ab =
2af =
2bg =
2ch =
4gh  2af = a^»y^j + aJj^yiiTi,
4hf  2bg = yi«e:jic, + yj^iTja?!,
4fg  2ch = gTi^j?,^, + z^Xjy^ ,
2bf = yi»y^, + ya^yigr^ ,
2cg = V^,a?a + 2rj9iria?i ,
2ah = iCi'^j + a;i*a?iyi,
2cf = Vya^i + ^a^^i ,
2ag = Xi^^ + x^ZiX^ ,
2bh = yi«aj,y8 + ya«a7iyi.
Proceeding next to the powers and products of the third order a', a'b, &c., the
total number of linear relations between the symmetric functions of the third degree
in respect to each set of roots exceeds by unity the number of the symmetric functions
of the form in question; in fact the expressions for abc, aP, bg*, ch', fgh, contain,
not five, but only four symmetric functions of the roots; for we have
abc = x^^Zy^ . x^iZ^y
4af» = {xjy^x^i + x^^x^z^) + ^lyiZiX^^^,
4bg» = (j/iZih/^^^ + j/j^aVi^O + ix^y^z^x^^^,
4ch* = {z^x^z^^ + z^cfzTjjf) + ^i^xZ^x^^^,
8fgh = {x^^x^i + x^}x^z^ ) '
+ {yxZxV^i + y%z^xx^)  + 2a^yi^i^a^j,
+ {Z^X^Z^} + Z^^ZtJI^ ) ,
C. II.
58
458 ON THE SYMMETBIO FUNCTIONS OF THB ROOTS [149
and consequently the quantities a, b, c, f, g, h, are not independent, but are connected
by the equation
abc  aP  bg«  ch« + 2fgh = 0,
an equation, which is in fact verified by the foregoing values of a, &c. in terms of
the coeflScients of the given sjrstem.
The expressions for the symmetric functions of the third degree considered as
Hinctions of a, b, c, f, g, h, are consequently not absolutely determinate, but they may
be modified by the addition of the term \ (abc — af* — bg* — ch' + 2fgh), where \ is an
indeterminate numerical coefficient.
The simplest expressions are those obtained by disregarding the preceding equation
for fgh, and the entire sjrstem then becomes :
at ^_ /M S/ft 3
^— U/i U/] ,
b' =»yiV»
b«c = yi'^iy,"^, ,
c^a ^ JB SC\Z^ cc^ f
bc» = VxZ^^f ,
ca" ^ZiX^z^,
ab« = xijf^^x^^,
abc s= ohyiZiX^tSt^
2a»f = x^^yiz^^ + x^y^xO^.
2b«g= y^z^x^}\y}z^^^,
2c»h = Zx^x^^^ + z^x^iZx\
2a fi ^~ x^ z^fjc^ "T x^ i9]M> ,
2b«h = yx^x^^^ + y2Si>{yx\
2c»f = Vy^,« + z^y,z,\
2a'h = x^z}x^ + x^z^Xy ,
2b«f = yi»a:,«yj + y,»a^«yi>
2c*g = z^%^z^ + z^h/^^Zx ,
2bcf = yx^z^^f + y}z^xz^ ,
2cag= z^XiZ^ +z^XiZiX^,
2abh= x^ix^^ ^ x^^iOxyx\
2bcg = yxz^x^^^ + y^ix^xZx ,
2cah = ZiX^x^^2 + z^x^iZ^ ,
2abf = x^j^x^^t + x^^xiy^z^ ,
149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 459
2bch=
VxZ^^H^t + yfz^^xZx ,
2caf =
z^x^x^^^ + zix^yyz^ ,
2abg=
x^%x^^^ + iJc^y^iyxZx.
4af»
2abc =
a?iy,V^j+a?j^aV^,
4bg»
2abc =
yiV^aVs + yV^q^y,,
4ch«
2abc =
Zx^y}Z% + Z^}\)^Z^ ,
4bf»
2b*c =
yxy^%^yiyxz^.
4cg»
2c«a =
z^z^i + z}z^x^ ,
4a.h«
2a«b =
x^x^^ + x}x^^.
4cP
2bc« =
Zx^%z^ + z^yx^z^ ,
4ag».
2ca« =
x^z^x^ + a^ V^ »
4bh«.
2ab» =
yx*^9%^yMY>
4agh
2a2f =
x^^x^^^ + x^x^y^z^.
4bhf
 2b«g =
yi*x^^^ h yj'x^iZi,
4cfg 
2c«h =
Z^^^% + ZfXTjj^Zy^ ,
4bgh'
 2abf =
yxZx^%y%^ y^z^y^.
4chf
 2bcg =
z^x^iz^ + z}x^^z^ ,
4afg >
2cah =
O^xZix^^ O^^^^x.
4cgh •
2acf =
yxz^xfz^ h y^ix^Zx ,
4ahf
2bag =
ZxX^}x^\ z^^y^x^.
4bfg
 2cbh =
i^xUxZ^y^^r x^fz^y^.
SPg
 4chf .
2bcg =
z^x^} + z^m^.
8g«h.
4a% 
2cah =
x^y^} + xiy^z^.
8h«f
 4bgh •
 2abf =
yxz^i + y^z^x^.
8fg»
4chg
2acf =
V^'ya + V^'yi,
8gh«.
4afh
 2bag =
a^yfz^^ x^y^z^,
8hf«.
4bgf .
 2cbh =
y^zix^^yiz^Xx,
8f» ■
 6bcf =
yxzi + y,V,
8g»
6cag =
z^x{ + z}x^y
8h» .
 6abh =
«i V + «a V
Secondly, consider the system of equations
(a, 6, c, /, flf, A, i, j, fc, fja?, y, ^)» = 0,
(a, /9, 7]^a?, y, ^) = 0,
58—2
460
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS
[149
where the cubic function written at full length is
Joining to the system the linear equation
(f I % ?$«. y, ^) = 0,
the linear equations give
and the resultant is
which may be represented by
(a, b, c, f, g. h, i, j, k, l$f , 17, ?)> = 0,
where the coefficients a, b, &c. are given by means of the Table : —
a
f
a =
b =
c =
3ya»
3ai8»
+ 3i8V
+ 87*0
+ 3a')3
2o/3y
f
f =
g =
b =
/a
+ /3V
a*
laJfiy
2a)3y
+ 2ya'
+ 20/3'
+ 2)8y'
2o»/3
2y*a
3«
3f«i,
•
1 =
•
J =
k=
A
o»6
+ a»
+ 2a^y
+ 2o/8y
+ 2aj8y
a/3»
+ ya'
ya«
+ /8y'
3{^ 1
3^
■
1 =
y««
«y3»
H
+ a»/3
+y3'y
+ 7*0
6M;
VIZ.
a = 67*  ciS*  3/i87» + 3ti8«7, &c.
But if the roots of the given system are
then the resultant of the three equations may also be represented by
and comparing with the former expression, we find :
a ^^ X\X^^y
b = yiy^a^j,
C s= Z\Z^^^
149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 461
3f = yiy^t + y^^x + y^iz^.
3g = ZiZ^x^ + z^z^ + z^iXi,
3i = y^z^t + ya«r^i + y^^z^,
3j = e^iiTa^s + Z^X>^ + ZgXiX^y
3k = ay^^, + iC8yayi+ a?,yiyj,
But there is in the present case a relation independent of the quantities a, &c., viz.
we have (a, )8, 7$iCi, yu ^i) = 0, (a, )8, y^x^, y,. 8^,) = 0, (a, $, 7$j^, y„ z,) = 0, and
thence eliminating the coefficients (a, /8, 7), we find
By forming the powers and products of the second degree a', ab, Ac, we obtain 55
equations between the symmetric functions of the second degree in each set of roots.
But we have V = = a symmetric ftmction of the roots, and thus the entire number
of linear relations is 56, and this is in fact the number of the synmietric functions
of the second degree in each set. I use for shortness the sign S to denote the sum
of the distinct terms obtained by permuting the different sets of roots, so that the
equations for the fundamental symmetric functions are —
a = ^^2^>
b= y,y^„
C = ZiZ^Zit
3f = S yij/j^s,
3g=S?i^2a?8,
Sh = SxjX^i,
3i =SyiZ^z^,
3j =SziX^3,
3k = S x^^t,
61 =Sx{y^^\
then the complete system of expressions for the symmetric functions of the second
order is as follows, viz.
a« = x^x^x^,
b» = yWyz\
c« = z^^z^H,\
be = yiz^y^^^t,
ca ^ Zi^XiZ^flc^ZfjX^f
ab= Xiy^x^^^t,
462 ON THE SYMMBTBIC FUNCTIONS OF THE BOOTS [l49
3af =^ S XitfiX^^^Xt,
3ch = S siiXiZ^x^^z^,
3bf =iSyi»y,»y,^„
3cg=iSfxV'8^s«8,
3ah = Sx^x^x^i,
3cf = S yiz,y^ztz^\
Ssig =^ S z,x,z,fc^\
Sbh = S x^ix^^^\
3ai  S x^^z^^z^j^,
3bj ^Sj/iz^x^^^^,
3ck = S ZiXiy^^iZs,
3bi = 8 yi^^^tz^,
3cj ^Szi^z^z^,
3ak = jS ari«a?3^jfl^„
3ci ^SyiZiZ^z^,
3aj =SziXiXi^Xi\
ahk = Sx^^^y^\
6al =Sxi^x^iZ^^,
6bl = iS yi*y^,a^„
6cl = S Zi^z^x^^z^,
9f«  6bi = S y,«y,V,
9g» 6cj =Sz,W^\
9h»  6ak = iS Xx"a:i«y,«,
9i'»  6cf = iS yl«^a«^,^
9j« 6ag = S^,»a:,%»,
9k«  6bh = S a?,«y,V.
9fg  3ck = iS Xjy^^^s\
9gh3ai ^Sy^z^z^^^\
9hf3bj ^SziXjX^^i^
149] OP CERTAIN SYSTEMS OF TWO EQUATIONS. 463
9jk  3af = Sxi^x^^j^i,
9ki  3bg = fif y^y^^z^^,
9ij — 3ch = S z^z^xi^^^,
9f i  3bc = S y,^y,z^,\
9gj — 8ca = S z^z^^x^y
9hk~ Sab = fif x^^x^^^,
3 ( Q + gk + hi  1«) = Sx^,z^x^^^,
3(2fj gkhi + l«) = fif^iy,a?4^^,«,
3(2gk hi fj +\*) = 8y,z,y,z^,\
3(2hi fj gk + l«) = flf^ia?,^^,y,«,
3(6fl 3ki.bg) =flfay^^,V,
3(6gl 3ijch) =8y,z,z,W,
3(6hl 3jkaf) =^Sz^x^x^^^\
3(6il 3fgck) =SziX^^%\
3(6jl 3ghai) ^Sx^,zW.
3(6kl3hfbj) ^Sy,z,x,%\
6(~fjgkhi + 41«) = fifa:i»y,V.
As an instance of the application of the formulae, let it be required to eliminate
the variables from the three equations,
(a, 6, c, /, g, h, i, j, A, i$a;, y, ^)» = 0,
(a\ b\ c\ /, ^, K Jx, y, zf = 0,
(a, yS, 7 $a;, y, 2^) = 0.
This may be done in two different ways; first, representing the roots of the linear
equation and the quadric equation by (a?i, yi, ^,), (a^, y,, z^, the resultant will be
(tt, ...$071, yi, ^,)».(a,...Ja;i, y,, erj)*,
which is equal to
a* aJj'^Tj' + &C.,
where the symmetric functions x^x^, &c. are given by the formulae a'* = ajj'a?,*, &c.,
in which, since the coefficients of the quadratic equation are {a\ h\ c\ f\ gf^ h'\
I have written a' instead of a. Next, if the roots of the linear equation and the cubic
equation are represented by {xi, yi, z^, (a?,, yj, z^, (x^, y,, ?,), then the resultant
will be
(tt', ...$da, yi. ^,)'.(a',...$d:2, ya, z^y(a\ ...^x^, y„ ^j^,
464 ON THE SYMMETRIC FUNCTIONS OF THE ROOTS &C. [149
which is equal to
a'* ayfxfx^ + &c.,
the symmetric functions x^^x^x^, &a being given by the formulaB d?=^x^x^x^, &c. The
expression for the Resultant is in each case of the right degree, viz. of the degrees
6, 3, 2, in the coefficients of the linear, the quadric, and the cubic equations respec
tively: the two expressions, therefore, can only differ by a numerical factor, which
might be determined without difficulty. The third expression for the resultant, viz.
(where (a?,, yi, Zi\,..(x^, y^, z^ are the roots of the cubic and quadratic equations)
compared with the foregoing value, leads to expressions for the fundamental sjrmmetric
functions of the cubic and quadratic equations, and thence to expressions for the other
symmetric functions of these two equations; but it would be difficult to obtain the
actually developed values even of the fundamental symmetric functiona I hope to
return to the subject, and consider in a general point of view the question of the
formation of the expressions for the other symmetric functions by means of the ex
pressions for the fundamental symmetric functiona
150]
465
150,
A MEMOIR ON THE CONDITIONS FOR THE EXISTENCE OF
GIVEN SYSTEMS OF EQUALITIES AMONG THE ROOTS OF
AN EQUATION.
[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for
the year 1857, pp. 727 — 731. Received December 18, 1856, — Read January 8, 1857.]
It is well known that there is a symmetric function of the roots of an equation,
viz. the product of the squares of the differences of the roots, which vanishes when any
two roots are put equal to each other, and that consequently such function expressed in
terms of the coefficients and equated to zero, gives the condition for the existence of a
pair of equal roots. And it was remarked long ago by Professor Sylvester, in some of
his earlier papers in the Philosophical Magazine, that the like method could be applied
to finding the conditions for the existence of other systems of equalities among the roots,
viz. that it was possible to form symmetric functions, each of them a sum of terms
containing the product of a certain number of the differences of the roots, and such that
the entire function might vanish for the particular system of equalities in question ;
and that such functions expressed in terms of the coefficients and equated to zero would
give the required conditions. The object of the present memoir is to extend this theory
and render it exhaustive, by showing how to form a series of types of all the different
functions which vanish for one or more systems of equalities among the roots; and in
particular to obtain by the method distinctive conditions for all the different systems of
equalities between the roots of a quartic or a quintic equation, viz. for each system con
ditions which are satisfied for the particular system, and are not satisfied for any other
systems, except, of course, the more special systems included in the particular system.
The question of finding the conditions for any particular system of equalities is essen
tially an indeterminate one, for given any set of functions which vanish, a function
syzygetically connected with these will also vanish; the discussion of the nature of the
C. II. 59
466 ON THE CONDITIONS FOR THE EXISTENCE OP GIVEN [150
syzygetic relations between the diflFerent functions which vanish for any particular
system of equalities, and of the order of the system composed of the several conditions
for the particular system of equalities, does not enter into the plan of the present
memoir. I have referred here to the indeterminateness of the question for the sake of
the remark that I have availed myself thereof, to express by means of invariants or
covariants the different systems of conditions obtained in the sequel of the memoir; the
expressions of the different invariants and covariants referred to are given in my 'Second
Memoir upon Quantics,' Philosophical TranscLCtions, vol. CXLVI. (1856), [141].
1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call
the roots a, 13, % S, e. Write 12 = 2<^(ai8)Ml2.13 = 2<^(ai8y(a7)"», 12.34 =
2^ (a — i8)'(7 — 8)**, &c., where ^ is an arbitrary function and i, m, &a are positive integers.
It is hardly necessary to remark that similar types, such as 12, 13, 45, &c., or as 12.13
and 23.25, &c., denote identically the same sums. Two types, such as 12.13 and
14.15.23.24.25.34.35.45, may be said to be complementary to each other. A par
ticular product (a — yS)(7 — 8) does or does not enter as a term (or factor of a term)
in one of the abovementioned sums, according as the type 12.34 of the product, or
some similar type, does or does not form part of the type of the sum; for instance, the
product (a — i8)(7 — S) is a term (or factor of a term) of each of the sums 12.34,
13.45.24, &c., but not of the sums 12.13.14.15, &a
2. If, now, we establish any equalities between the roots, e.g. ci = l3, 7 = 8, the
effect will be to reduce certain of the sums to zero, and it is easy to find in what
cases this happens. The sum will vanish if each term contains one or both of the &ctors
^"ffi 7— Si i.e. if there is no term the complementary of which contains the product
(a — yS) (7 — 8), or what is the same thing, whenever the complementary type does not con
tain as part of it, a type such as 12.34. Thus for the sum 14.15.24.25.34.35.45,
the complementary type is 12.13.23, which does not contain any type such as 12.34,
i. e. the sum 14.15.24.25.34.35.45 vanishes for a = y3, 7 = 8. It is of course clear
that it also vanishes for a = yS = e, 7 = 8 or a = yS = 7 = 8, &a, which are included in
^ = fi» 7=8. But the like reasoning shows, and it is important to notice, that the
sum in question does not vanish for a = ^ = y: and of course it does not vanish for
a = ^. Hence the vanishing of the sum 14.15.24.25.34.35.45 is characteristic of the
system a = yS, 7=8. A system of roots a, yS, 7, 8, e may be denoted by 11111; but
if a = y3, then the system may be denoted by 2111, or if a = i8, 7 = 8, by 221, and
so on. We may then say that the sum 14.15.24.25.34.35.45 does not vanish for
2111, vanishes for 221, does not vanish for 311, vanishes for 32, 41, 5.
3. For the purpose of obtaining the entire system of results it is only necessary to
form Tables, such as the annexed Tables, the meaning of which is sufficiently explained
by what precedes: the mark (x) set against a type denotes that the sum represented
by the complementary type vanishes, the mark (o) that the complementary type does
not vanish, for the system of roots denoted by the symbol at the top or bottom of the
column; the complementary type is given in the same horizontal line with the original
type. It will be noticed that the righthand columns do not extend to the foot of the
Table ; the reason of this of course is, to avoid a repetition of the same type. Some of
BTSTEH8 OF EQUAUTIES AUONQ THE ROOTS OF AN EQUATION.
467
150]
the types at the foot of the Tables are complemeDtaiy to themselves, but I have, not
withstauding this, given the complementary type in the form under which it naturally
presents itself.
4. The Tahles are:
Table for the equal Boots of a Quartic.
211
22
31
4
14 . 23 . 24 . 34
o
o
■s'
o
14 . 23 . S4 . 34
o
o
o
X
14 . 23 . 24 . 34
o
o
o
»
13 . 14 . 23 . 24
a
o
X
X
"IT 23 . 24 . 34
—
—
—
84 14 . 23 . 34
sn
22
31
4
23 14 . 24 . 34
Table for the equal Boots of a Quintia
Sill
221
311
32
*1
fi
~^
T
7
o
X
X
X
X X
o
X
X
X
X X
o
o
X
X
X ly
o
X
x
X
X ix
o
o
X
X
xlx
o
X
X
X jx
o
y.
o
y
X x
o
X
X
X
X X
o
o
a
X
X >
o
o
X
X
X X
o
X
X
X
X
O
X
X
X
X
o
X
X
o
o
o
X
X
X
o
o
X
X
X
o
o
o
X
X
X
o
o
X
X
X
o
a
o
o
X
X
o
o
X
X
X
^
Bill
ssT
311
32
41
7
2111
2«
51
32
41
b
35 . 46
o
~~^
~^
a
35 . 45
a
X
35 . 45
o
X
35 . 45
o
a
X
X
36 . 46
o
o
a
y
34 . 35
o
o
X
X
35 . 46
o
o
X
X
35 . 46
o
o
X
X
X
35 . 45
o
o
X
X
36 . 46
o
o
X
X
X
36 . 46
o
X
X
X
36 . 46
o
o
X
X
34 . 45
o
o
X
X
34 . 36
o
o
X
X
X
X
35 . 45
35 . 45
2111
221
311
32
41
b
36 . 45
'
35 . 46
34 . 36
26 . 36
The two Tables enable the discussion of the theory of the equal roots of a quartic or
quintic equation: first for the quartic:
5. In order that a quartic may have a pair of equal roots, or what is the same
thing, that the system of roots may be of the form 211, the type to be considered is
12.13.14.23.24.34;
59—2
468 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN [l50
this of course gives as the function to be equated to zero, the discriminant of the
quartic.
6. In order that there may be two pairs of equal roots, or that the system may
be of the form 22, the simplest type to be considered is
14.24.34;
this gives the function
which being a covariant of the degree 3 in the coefficients and the degree 6 in the
variables, can only be the cubicovariant of the quartic.
7. In order that the quartic may have three equal roots, or that the system of
roots may be of the form 31, we may consider the type
13.14.23.24,
and we obtain thence the two functions
2(a7)(aS)(iS7)(^S),
2(a7)«(a8)(^7)(/38)',
which being respectively invariants of the degrees 2 and 3, are of course the quadrin
variant and the cubinvariant of the quartic. If we had considered the apparently more
simple type
12.34,
this gives the function
which is the quadrivariant, but the cubinvariant is not included under the type in
question.
8. Finally, if the roots are all equal, or the system of roots is of the form 4, then
the simplest type is
12;
and this gives the function
a covariant of the degree 2 in the coefficients and the degree 4 in the variables ; this is
of course the Hessian of the quartic.
Considering next the case of the quintic:
9. In order that a quintic may have a pair of equal roots, or what is the same
thing, that the system of roots may be of the form 2111, the type to be considered is
12.13.14.15.23.24.25.34.35.45;
this of course gives as the function to be equated to zero, the discriminant of the
quintic.
150] SYSTEMS OP EQUALITIES AMONG THE ROOTS OF EQUATION. 469
10. In order that the quintic may have two pairs of equal roots, or that the
system of roots may be 221, the simplest type to be considered is
14.15.24.26.34.35.45;
a type which gives the function
2 (a  S)(a  e)(^  8)(yS  e)(7  S)(7  e)(B  €)« (a;  ay)' (x  ySy)« (x  7y)».
This is a covariant of the degree 5 in the coefiBcients and of the degree 9 in the variables ;
but it appears from the memoir above referred to, that there is not any irreducible
covariant of the form in question; such covariant must be a sum of the products
(No. 13)(No. 20), (No. 13)(No. 14)», (No. 15)(No. 16) (the numbers refer to the Cova
riant Tables given in the memoir), each multiplied by a merely numerical coefficient.
These numerical coefficients may be determined by the consideration that there being
two pairs of equal roots, we may by a linear transformation make these roots 0, 0, oo , oo ,
or what is the same thing, we may write a = 6 = e=/=0, the covariant must then
vanish identically. The coefficients are thus found to be 1, — 4, 50, and we have for a
covariant vanishing in the case of two pairs of equal roots,
1 (No. 13)(No. 20)
 4 (No. 13)(No. Uy
+ 50 (No. 15)(No. 16)
[or in the new notation AH — ^AB* + 50CD].
In fact, writing a = 6 = e=/=0, and rejecting, where it occurs, a factor ic'y*, the several
covariants become functions of ex, di/; and putting, for shortness, x, y instead of ex, dy,
the equation to be verified is
1 . 10(a: + y)(6a:* + Sa^y + 28«y + Sxy^ + 6y*)
 4i.lO{x'\y){Za? + %xy + Zr/'y
+ 50(6a;* + &ry + 6y»)(a^ + ic'y + a?y« + y») = ;
and dividing out by {x + y) and reducing, the equation is at once seen to be identically
true.
11. In order that the quintic may have three equal roots, or that the system
of roots may be of the form 311, the simplest type to be considered is
12.13.23.45;
this gives the function
2(ayS)>(i8«7)'(7«)'(S«)*,
which being an invariant, and being of the fourth degree in the coefficients, must be
the quartinvariant of the quintic [that is No. 19, = 0\ The same type gives also the
function
2(a^)«(i87)«(7a)«(S.6)«(a:.Sy)»(a:.6y)»,
470 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN SYSTEMS, &C. [150
which is a covariant of the degree 4 in the coefficients and the degree 4 in the
variables; and it must vanish when a = 6 = c = 0, this can only be the covariant
3 (No. 20) 2 (No. 14)>, [=3ir25«],
which it is clear vanishes as required.
12. In order that the quintic may have three equal roots and two equal roots,
or that the system of roots may be of the form 32, the simplest type to be con
sidered is
12.13.14.15,
which gives the function
2(a^)(a7)(aS)(ae)(a;ySy)«(^7y)*(a?Sy)«(^6y)«,
a covariant of the degree 4 in the coefficients, and the degree 12 in the variables;
and it must vanish when a = 6 = c = 0, e=/=0; this can only be the covariant
3 (No. 13)»(No. 14) 25 (No. 15)», [= 3^»B  25C*],
which it is clear vanishes as required.
13. In order that the quintic may have four equal roots, or that the system
may be of the form 41, the simplest type to be considered is
12.34,
which gives the function
2(ayS)«(7S)'(^6y)»,
a covariant of the degree 2 in the coefficients, and of the same degree in the variables;
this can only be the covariant (No. 14), [=B].
14. Finally, in order that all the roots may be equal, or that the system of
roots may be of the form 5, the type to be considered is
12;
and this gives the function
a covariant of the degree 2 in the coefficients, and the degree 6 in the variables,
and this can only be the Hessian (No. 15), [=s C].
It will be observed that all the preceding conditions are distinctive; for instance,
the covariant which vanishes when the system of roots is of the form 311, does not
vanish when the system is of the form 221, or of any other form not included in
the form 311.
151]
471
151.
TABLES OF THE STUEMIAN FUNCTIONS FOR EQUATIONS OF
THE SECOND, THIRD, FOURTH, AND FIFTH DEGREES.
[From the Philosophical Transdctiona of the Royal Society of London, vol. CXLVII. for
the year 1857, pp. 733—736. Received December 18, 1856,— Read January 8, 1857.]
The general expressions for the Sturmian functions in the form of determinants
are at once deducible firom the researches of Professor Sylvester in his early papers
on the subject in the Philosophical Magazine, and in giving these expressions in the
Memoir 'Nouvelles Recherches sur les Fonctions de M. Sturm,' Liouville, t. xiii. p. 269
(1848), [65], I was wrong in claiming for them any novelty. The expressions in the
lastmentioned memoir admit of a modification by which their form is rendered some
what more elegant; I propose on the present occasion merely to give this modified
form of the general expression, and to give the developed expressions of the functions
in question for equations of the degrees two, three, foUr, and five.
Consider in general the equation
Cr = (a, 6, ... j, k^x, 1)»,
and write
P = (a, 6, ... jJix, ir'\
Q=(6, ... j^fcJix, l)\
then supposing as usual that the first coefficient a is positive, and taking for shortness
th •"" 1 w """ 1 fh "■" 2
?ij, n,, &c. to represent the binomial coefficients — = — , :.—^ , &c. corresponding
to the index (n — 1), the Sturmian functions, each with its proper sign, are as
follows, viz.
472
TABLES OF THE STURMIAN FUNCTIOKS FOR
ri5i
U. P.
P. Q
a , b
 *P. P.
'a. • ,
iij6, a ,
*«, Q
6, .
fliC^ b i
I
+ a^P, xP, P. a»e. xQ. Q . &c
a.
I
a,
njb.
a,
ry:.
6.
6. .
w,c, 6
where the terms contaimng the powers of x, which exceed the degrees of the several
functions respectively, vanish identically (as is in fact obvious from the form of the
expressions), but these terms may of course be omitted ab initio.
The following are the results which I have obtained; it is well known that the
last or constant function is in each case equal to the discriminant, and as the
expressions for the discriminant of equations of the fourth and fifth degrees are given,
Tables No. 12 and No. 26 [Q', see 143] in my 'Second Memoir upon Quantics'(0. I
have thought it sufficient to refer to these values without repeating them at length.
Table for the degree 2.
The Sturmian functions for the quadric (a, 6, c$a?, 1)* are
c + 1 \l^x, ly.
(! a+ I 6 + 1
5*. 1).
ac—\
6» + l
Table for the degree 3.
The Sturmian functions for the cubic (a, 6, c, d$a?, ly are
a+l
b + 3 e + 3 (i+1
5*^ 1)*.
1 Philotophieal Trantactions, t. cxlvi. p. 101 (1856), [141].
151] EQUATIONS OF THE SECOND, THIRD, FOURTH, AND FIFTH DEGREES. 473
a + 1
6 + 2
c+l
'$.«>, 1)',
$«. 1).
1 a'cP
+ 1
abed
+ 6
' ac"
4
' bd"
4
b'c^
3
Table for the degree 4.
The Sturmian functions for the quartic (a, 6, c, d, e$x, 1\ are
(
a+ 1
6 + 4
c + 6
(^ + 4
6 + 1
5*. 1)*'
a+ 1
6 + 3
c + 3
c^+l
$*. 1)'.
ac— 3
6^+3
ac;3
6c +3
ae\
bd^l
Ja^. 1)'.
3(
a^ce  1
a^cfe + 1
a»c;» + 3
a6cc —4
ah^e + 1
ahd:" 1
o6c€? — 14
ac^d +3
ac» + 9
6»e +3
6»c^ + 8
6»cc; 2
6V  6
$«^ 1),
aV+1
Disct. Tab.
No. 12.
Table for the degree 5.
The Sturmian functions for the quintic (a, 6, c, d, e, /$a:, 1)" are
a + 1
6 + 5
c+10
ef+10
e + 5
/+i r^x, i)»,
C. II.
60
474
TABLES OF THE 8TUEM1AN FUNCTIONS, &C.
[151
a + 1
6 + 4
c + 6
dh4
e+l
Jix, ly,
(
oc  4
6« +4
arf6
6c +6
a« — 4
bd+i
a/l
6« +1
"ga:, 1)',
2(
a«ce  8
a V  2
a»c(/* + 3
a>^ +18
a«^ +12
a6c/ 11
a6»c + a
a6y+ 2
abde 3
a6crf 76
abce  42
ac"e + 8
ac» +48
a6rf« 12
6y + 8
b^d +40
cuM +32
6«cc  5
6V 30
6»e + 30
b^cd  20
\', 1)",
2(
aV" 
2
a^dp +
3
a^def +
24
aV/ 
8
aV 
32
a^bcP 
11
a>6y> +
2
a^bdef +
58
a»Me* +
264
a»6c» +
8
a'ftcc/" —
52
aV«/* +
104
a^bd}/
96
a^ccP/ 
156
a^(?df +
64
a«ccfe« 
96
aVtf> +
352
a*cPe +
108
a'ccPe 
938
a6y* +•
8
a»(£* +
432
a6*c«/ 
266
aJt^ef +
28
ab^d^f 
8
oftW 
970
ah^d^ +
35
o^'cPe +
120
abi^df ^
584
abi^de +
2480
abi?^ +
120
aJl^cdf ^
264
a6cc?e 
360
ahcd^ 
1440
oc*/ 
288
a6cy 
192
W?dA +
160
ac*« 
960
6V +
120
cu^d^ +
640
6»ai/ 
320
6*(^ 
160
6»ce« 
75
6V +
450
6»(?6 +
200
l^cde 
1400
6»cy +
180
6»d» +
800
6«c»^ 
100
6V/ +
120
6V6 +
600
6Vrf» 
400
aV* + l
+ Ac,
Disct Tab.
No. 26, [Q'].
5^, 1),
152]
475
152,
A MEMOIR ON THE THEORY OF MATRICES.
[From the Philosophical Transactions of the Royal Society of London, vol. CXLVili. for
the year, 1858, pp. 17 — 37. Received December 10, 1857, — Read January 14, 1858.]
The term matrix might be used in a more general sense, but in the present
memoir I consider only square and rectangular matrices, and the term matrix used
without qualification is to be understood as meaning a square matrix ; in this restricted
sense, a set of quantities arranged in the form of a square, e.g.
( a , b , c )
a' , V , d
a", V\ d'
is said to be a matrix. The notion of such a matrix arises naturally firom an
abbreviated notation for a set of linear equations, viz. the equations
z =
ax + 6y '\cz ,
Y^a'x + Vy + dz ,
Z ^(j^'x^V'y^c'z,
may be more simply represented by
(Z, F, ^ = ( a , 6 , c '^x, y, «),
a' , V , d
a", V\ d'
and the consideration of such a system of equations leads to most of the fundamental
notions in the theory of matrices. It will be seen that matrices (attending only to
those of the same order) comport themselves as single quantities; they may be added,
60—2
476
A MEMOIB ON THE THEORY OF MATRICES.
[152
multiplied or compounded together, &c. : the law of the addition of matrices is pre
cisely similar to that for the addition of ordinar}' algebraical quantities; as regards
their multiplication (or composition), there is the peculiarity that matrices are not in
general convertible; it is nevertheless possible to form the powers (positive or negative,
integral or fractional) of a matrix, and thence to arrive at the notion of a rational
and integral function, or generally of any algebraical function, of a matrix. I obtain
the remarkable theorem that any matrix whatever satisfies an algebraical equation of
its own order, the coeflBcient of the highest power being unity, and those of the
other powers functions of the terms of the matrix, the last coefiBcient being in bet
the determinant; the rule for the formation of this equation may be stated in the
following condensed form, which will be intelligible after a perusal of the memoir,
viz. the determinant, formed out of the matrix diminished by the matrix considered
as a single quantity involving the matrix unity, will be equal to zero. The theorem
shows that every rational and integral function (or indeed every rational function) of
a matrix may be considered as a rational and integral function, the degree of which
is at most equal to that of the matrix, less unity; it even shows that in a sense,
the same is true with respect to any algebraical function whatever of a matrix. One
of the applications of the theorem is the finding of the general expression of the
matrices which are convertible with a given matrix. The theory of rectangular
matrices appears much less important than that of square matrices, and I have not
entered into it further than by showing how some of the notions applicable to these
may be extended to rectangular matrices.
1. For conciseness, the matrices written down at full length will in general be
of the order 3, but it is to be understood that the definitions, reasonings, and con
clusions apply to matrices of any degree whatever. And when two or more matrices
are spoken of in connexion with each other, it is always implied (unless the contrary
is expressed) that the matrices are of the same order.
2. The notation
( a , 6 , c ^x, y, z)
a', 6', c'
represents the set of linear functions
((a, 6, c\x, y, z\ (a', h\ d\x, y, z\ (a", 6", c"\x, y, z)\
so that calling these (X, F, Z), we have
(X, F, ^ = ( a , 6 , c \x, y, z)
and, as remarked above, this formula leads to most of the fundamental notions in the
theory.
152]
A MEMOIR ON THE THEORY OF MATRICES.
477
3. The quantities {X, Y, Z) will be identically zero, if all the terms of the matrix
are zero, and we may say that
( 0, 0, .)
0, 0,
0, 0,
is the matrix zero.
Again, {X, T, Z) will be identically equal to {x, y, z), if the matrix is
(1. 0. )
0, 1. !
0, 0, 1 :
and this is said to be the matrix unity. We may of course, when for distinctness it
is required, say, the matrix zero, or (as the case may be) the matrix unity of such an
order. The matrix zero may for the most part be represented simply by 0, and the
matrix unity by 1.
4. The equations
(Z, F, Z) = { a, b, c Jix.y. z\ X\ F, Z') = ( « , /S , 7 $.^, y, z)
a' , 6' , c'
^" J." V
a , , c
«'. /3', 7'
a", /8", 7"
give
{X + X, Y+T, Z + Z) = { a +a , b +0 , c +y Jix, y, z)
a' +a', V +yS', c'+7'
a" + a", 6" + ^", (Z' + y
and this leads to
(o+a, b +P , c+7 ) = (a, b , c ) + (o, fi , 7 )
o' +0', b' +ff, c' +7'
o" + a", 6"+/3", c"+7"
a' , b' , c'
a", b", c"
as a rule for the addition of matrices ; that for their subtraction is of course similar
to it.
5. A matrix is not altered by the addition or subtraction of the matrix zero,
that is, we have M ±0 = M.
The equation L==M, which expresses that the matrices Z, M are equal, may also
be written in the form ZcrM^^O, Le. the difference of two equal matrices is the
matrix zero.
6. The equation L^My written in the form Z + Jf^O, expresses that the sum
of the matrices L, M \a equal to the matrix zero, the matrices so related are said to be
opposite to each other ; in other words, a matrix the terms of which are equal but oppo
site in sign to the terms of a given matrix, is said to be opposite to the given matrix.
478
A MEMOIR ON THE THEORY OF MATRICES.
[152
7. It is clear that we have L {M = M + L, that is, the operation of addition is
commutative, and moreover that (L + M) + JV = Z + (M iIf^^L + M + N, that is, the
operation of addition is also associative.
8. The equation
written under the forms
(X, Y, Z) — { a , 6 , c $ww?, my, mz)
(X, T, Z) = m( a , 6 , c $a:, y, z) = ( via , mb , mc ^x, y, z)
a\ 6' , c'
a , , c
ma' , mi/ , mc'
ma", fii6", m&'
gives
m( a y 6, c ) = ( ma , mh , mc )
a\ h\ c'
a'\ h'\ c"
tna' , mV , mc'
?7ia", m6", mc"
as the rule for the multiplication of a matrix by a single quantity. The multiplier m
may be written either before or after the matrix, and the operation is therefore com
mutative. We have it is clear m{L\M) — mL + mJf, or the operation is distributive.
9. The matrices L and mL may be said to be similar to each other; in
particular, if m = 1, they are equal, and if m = — 1, they are opposite.
10. We have, in particular,
m ( 1, 0, ) = ( m, 0, ),
0, m,
0, 0, m
0, 1,
0, 0, 1
or replacing the matrix on the lefthand side by unity, we may write
m=={m, 0, );
0, m,
0, 0, m
the matrix on the righthand side is said to be the single quantity m considered as
involving the matrix unity.
11. The equations
(Z, F, Z) = ( a,
a ,
6 , c $a?, y, z\ (a?, y, ^) = ( a , ^ , 7 $f, 1;, {),
6', C a', /3', y
i", c" a", /3", y
152]
A MEMOIR ON THE THEORY OF MATRICES.
479
give
(X,7.Z) = (A, B, C IJf , 17. ?) = ( o , 6, c ^a.
A', B, C
A", B", C"
a'. V,
a , 6 ,
jf
a,
fjt Q>l ^11
and thence, substituting for the matrix
{A , B , C )
A', R. C
I A", F', C"
its value, we obtain
{{a.b.c^a. a', a"), {a,b ,c^P, &, /8"). (a , 6 . c 57, 7. 7") )  ( « . & . c $ a . ^ . 7 )
(a' .b'.c' $a, a', O, (a' . 6' , c' 3[y9, ff, /3"), (a' , 6' . c' $7, 7'. 7")
(a", 6", c"$a, a', a"), (a". 6", c"$/8, ff, n. (o", 6". c"$7. 7'. 7")
o',6',c'
_// 1;// ^
«',/3'.7'
a", /8". 7"
as the rule for the multiplication or composition of two matrices. It is to be
observed, that the operation is not a commutative one; the component matrices may
be distinguished as the first or further component matrix, and the second or nearer
component matrix, and the rule of composition is as follows, viz. any line of the com
pound matrix is obtained by combining the corresponding line of the first or further
component matrix successively with the several columns of the second or nearer com
pound matrix.
[We may conveniently write
(a. «', a"). (A /9'. ^'). (7. 7'. 7")
(a ,
b.
c)
if
f>
ij
(a',
b',
c')
t>
it
if
(0".
b".
c")
n
»
n
to denote the lefthand side of the last preceding equation.]
12. A matrix compounded, either as first or second component matrix, with the
matrix zero, gives the matrix zero. The case where any of the terms of the given
matrix are infinite is of course excluded.
13. A matrix is not altered by its composition, either as first or second component
matrix, with the matrix unity. It is compounded either as first or second component
matrix, with the single quantity m considered as involving the matrix unity, by
multiplication of all its terms by the quantity m:. this is in fact the beforementioned
rule for the multiplication of a matrix by a single quantity, which rule is thus seen
to be a particular case of that for the multiplication of two matrices.
14. We may in like manner multiply or compound together three or more
matrices: the order of arrangement of the factors is of course material, and we may
480
A MEMOIR ON THE THEORY OF MATRICES.
[152
distinguish them as the first or furthest, second, third, &c., and last or nearest
component matrices: any two consecutive factors may be compounded together and
replaced by a single matrix, and so on until all the matrices are compound^ together,
the result being independent of the particular mode in which the composition is
effected; that is, we have L.MN^LM .N ^ LMN, LM.NP^L. MN.P, &a, or the
operation of multiplication, although, as already remarked, not commutative, is associative.
15. We thus arrive at the notion of a positive and integer power 2> of a
matrix L, and it is to be observed that the different powers of the same matrix are con
vertible. It is clear also that p and q being positive integers, we have L^.L^^L^^^,
which is the theorem of indices for positive integer powers of a matrix.
16. The lastmentioned equation, 2>.Z« = 2>+«, assumed to be true for all values
whatever of the indices p and q, leads to the notion of the powers of a matrix for any
form whatever of the index. In particular, 2> . Z® = J!> or Z<* = 1, that is, the 0th power
of a matrix is the matrix unity. And then putting p = l, } = — 1, or /> = — !, g=l, we
have L . L"^ = Z~^ . Z = 1 ; that is, L~\ or as it may be termed the inverse or reciprocal
matrix, is a matrix which, compounded either as first or second component matrix
with the original matrix, gives the matrix unity.
17. We may arrive at the notion of the inverse or reciprocal matrix, directly
from the equation
(Z, F. Z) = ( a , 6 , c Jxy y, z\
a' , 6' , c'
«" Uf V
a , by c
in fact this equation gives
{XyyyZ) = (A, A\ ^"$Z, F, Z) = (( a, 6, c )^$Jr, F, Z),
5, B, B'
a , , c
and we have, for the determination of the coefficients of the inverse
matrix, the equations
(A. A', ^"$o, 6, c ) = ( 1. 0, ),
or reciprocal
B, ff, B'
C, C, C"
a', V, c'
a", b", c"
0. 1,
0, 0, 1
(a . b, c 11 A, A', A" ) = ( 1, 0, ),
a' , 6' , c'
a ,
6",
.//
B, B, B'
C, C, C"
0, 1,
0, 0. 1
s
152]
A MEMOIR ON THE THEORY OF MATRICES.
481
which are equivalent to each other, and either of them is by itself sufficient for the
complete determination of the inverse or reciprocal matrix. It is well known that if
V denote the determinant, that is, if
V = a , 6 , c
a' , 6' , c'
^" J." «"
a , , c
then the terms of the inverse or reciprocal matrix are given by the equations
V
1, 0,
0, b', c'
0, h". c"
B = l
V
, 1,
a' , 0, c'
a", 0, c"
, &c.
or what is the same thing, the inverse or reciprocal matrix is given by the equation
( a , 6 , c )^ 1 ( BaV, Ba'V, a„^ V )
a', b\ c'
SfrV, a,,.v, a^v
ScV, ac'V, a,..v
where of course the differentiations must in every case be performed as if the terms
a, 6, &c. were all of them independent arbitrary quantities.
18. The formula shows, what is indeed clear d priori, that the notion of the
inverse or reciprocal matrix fails altogether when the determinant vanishes: the matrix
is in this case said to be indeterminate, and it must be understood that in the
absence of express mention, the particular case in question is frequently excluded from
consideration. It may be added that the matrix zero is indeterminate; and that the
product of two matrices may be zero, without either of the factors being zero, if only
the matrices are one or both of them indeterminate.
19. The notion of the inverse or reciprocal matrix once established, the other
negative integer powers of the original matrix are positive integer powers of the
inverse or reciprocal matrix, and the theory of such negative integer powers may be
taken to be known. The theory of the fractional powers of a matrix will be ftirther
discussed in the sequel.
20. The positive integer power L^ of the matrix L may of course be multiplied
by any matrix of the same degree: such multiplier, however, is not in general con
vertible with L; and to preserve as far as possible the analogy with ordinary
algebraical frmctions, we may restrict the attention to the case where the multiplier
is a single quantity, and such convertibility consequently exista We have in this
manner a matrix cL^, and by the addition of any number of such terms we obtain
a rational and integral function of the matrix L.
a 11. 61
482 A MEMOIR ON THE THEORY OF MATRICES. [132
21. The general theorem before referred to will be best anderstood by a com
plete deTelopment of a particular case. Imagine a matrix
JI={ a, 6 ),
c, d
and form the determinant
a — M, h
c ,dM
the developed expression of this determinant is
Jf »  ( a + rf) JT + (ad  6c) Jf • ;
the values of M\ M\ Jf* are
( a^^hc , 6(a + rf) ), ( a. 6 ). ( 1. X
I ' i
cia^d), (f + 6c ' c, d\ ! 0, 1 i
and substituting these values the determinant becomes equal to the matrix zero, vis.
we have
aJf, 6 =(a» + ftc, 6(a + ci) )(a + rf) ( a, 6 ) + (adftc) ( 1, 0)
c ,dM\ I c(a + d), cf + ftc ! c, d ^ 0, 1 I
= ( (a» + ftc)(a + ci)a + (a(i6c), 6(a + d)(a + rf)6 ) = ( 0, );
c(a + d)(a + d)c , cf + ftc(a + cOd+arf6c ! ! 0, I
that is
a  Jf , 6 . = 0,
c ,d^M
where the matrix of the determinant is
( a, 6 )i/( 1, ),
c, d I I 0, 1 I
that is, it is the original matrix, diminished by the same matrix considered as a single
quantity involving the matrix unity. And this is the general theorem, viz. the deter
minant, having for its matrix a given matrix less the same matrix considered as a
single quantity involving the matrix unity, is equal to zero.
22. The following symbolical representation of the theorem is, I think, worth
noticing: let the matrix if, considered as a single quantity, be represented by JB, then
writing 1 to denote the matrix unity, S,l will represent the matrix M, considered
as a single quantity involving the matrix unity. Upon the like principles of notation,
I. if will represent, or may be considered as representing, simply the matrix if, and
the theorem is
Det (I.ifiBf.l) = 0.
152]
A MEMOIR ON THE THEORY OF MATRICES.
483
23. I have verified the theorem, in the next simplest case of a matrix of the
order 3, viz. if Jf be such a matrix, suppose
if = ( a, 6, c ),
d, e, f
then the derived determinant vanishes, or we have
a — M, b
d , e
9 , h
> c
M, f
, iif
= 0,
or expanding
Jf* — (a + e + i) if ^ + (^' + ta + 06 —/A — eg — bd) M — {ad + bfg + cdh — afh — bdi — ceg) = ;
but I have not thought it necessary to undertake the labour of a formal proof of
the theorem in the general case of a matrix of any degree.
24. If we attend only to the general form of the result, we see that any matrix
whatever satisfies an algebraical equation of its own order, which is in many cases the
material part of the theorem.
25. It follows at once that every rational and integral function, or indeed every
rational function of a matrix, can be expressed as a rational and integral function of
an order at most equal to that of the matrix, less unity. But it is important to
consider how far or in what sense the like theorem is true with respect to irrational
functions of a matrix. If we had only the equation satisfied by the matrix itself,
such extension could not be made; but we have besides the equation of the same
order satisfied by the irrational function of the matrix, and by means of these two
equations, and* the equation by which the irrational function of the matrix is deter
mined, we may express the irrational function as a rational and integral function of
the matrix, of an order equal at most to that of the matrix, less unity; such expression
will however involve the coefficient of the equation satisfied by the irrational function,
which are functions (in number equal to the order of the matrix) of the terms,
assumed to be unknown, of the irrational function itself. The transformation is never
theless an important one, as reducing the number of unknown quantities firom n' (if n
be the order of the matrix) down to n. To complete the solution, it is necessary to
compare the value obtained as above, with the assumed value of the irrational function,
which will lead to equations for the determination of the n unknown quantitiea
26. As an illustration, consider the given matrix
M=( a, 6 ),
c, d
61—2
484
A MEMOIR ON THE THEORY OF MATRICES.
[152
and let it be required to find the matrix L = *JM, In this case M satisfies the equation
and in like manner if
then L satisfies the equation
if»(a + d)Jf + ad6c = 0;
7. S
Z«  (a + S) Z + aS /87 = ;
and from these two equations, and the rationalized equation Z' = M, it should be possible
to express L in the form of a linear function of M: in fact, putting in the last
equation for Z" its value (=M), we find at once
Z =
ig[.lf + (aS^7)],
which is the required expression, involving as it should do the coefficients a + S, aS^/3y
of the equation in L. There is no difficulty in completing the solution; write for
shortness a + S = X, aS — ySy = F, then we have
Z = ( 0, fi ) = ( a'\Y
X
b_ ),
X
d+T
and consequently forming the values of a + S and aS — ^87,
X =
a+d+2Y
y_ (a+r)(d+r)6c
and putting also a + d = P, ad — bc^Q, we find without difficulty
and the values of a, /8, 7, S are consequently known. The sign of ^Q is the same in
both formulae, and there are consequently in all four solutions, that is, the radical vif
has four values.
27. To illustrate this further, suppose that instead of M we have the matrix
if> = ( a, 6 )» = ( d' + bc , 6(a + d) ),
c, d
c(a + d), d» + 6c
152]
A MEMOIR ON THE THEORY OF MATRICES.
485
so that D = M^, we find
P = (aHd)«2(ad6c),
and thence VQ = ± (ad — be). Taking the positive sign, we have
and these values give simply
But taking the negative sign,
F= ad — be,
X=±(a + d),
L=±^( a, b )=±M,
I c, d
Y=ad + bCy
X = ± V(a  d)« + 46c,
and retaining X to denote this radical, we find
Z =
_( a'ad+26c b(a + d) ),
X
c(a'{d)
X
d^ad + 2bc
which may also be written
, a+d(a, b) 2(ad6c) ( 1, ),
^ I 0, 1 I
or, what is the same thing.
c, d
J ^a + d ^ 2 (ad — be)
" Z X '
and it is easy to verify d posteriori that this value in fact gives D = M\ It may
be remarked that if
if» = ( 1, )» = 1.
0, 1
the lastmentioned formula fails, for we have X = ; it will be seen presently that
the equation Z* = 1 admits of other solutions besides Z = ± 1. The example shows how
the values of the fractional powers of a matrix are to be investigated.
28. There is an apparent difficulty connected with the equation satisfied by a
matrix, which it is proper to explain. Suppose, as before,
M^( a, 6 ),
c, d
486
A MEMOIB ON THE THEORY OF MATRICES.
[152
so that M satisfies the equation
a  If, 6 =0,
and let X^, X„ be the single quantities, roots of the equation
or
c d — X
=
or
Z«(a+(i)Z + ad6c=0.
The equation satisfied by the matrix may be written
(ifZ,)(ifZ„) = 0.
in which X,, X,, are to be considered as respectively involving the matrix unity, and it
would at first sight seem that we ought to have one of the simple fiEu^tors equal to
zero; this is obviously not the case, for such equation would signify that the perfectly
indeterminate matrix M was equal to a single quantity, considered as involving the
matrix unity. The explanation is that each of the simple factors is an indeterminate
matrix, in fact M — X, stands for the matrix
(aX„
b
dX
).
and the determinant of this matrix is equal to zero. The product of the two factors
is thus equal to zero without either of the factors being equal to zero.
29. A matrix satisfies, we have seen, an equation of its own order, involving the
coefficients of the matrix; assume that the matrix is to be determined to satisfy some
other equation, the coefficients of which are given single quantities. It would at first
sight appear that we might eliminate the matrix between the two equations, and thus
obtain an equation which would be the only condition to be satisfied by the terms
of the matrix ; this is obviously wrong, for more conditions must be requisite, and we
see that if we were then to proceed to complete the solution by finding the value of
the matrix common to the two equations, we should find the matrix equal in every case
to a single quantity considered as involving the matrix unity, which it is clear ought
not to be the case. The explanation is similar to that of the difficulty before adverted
to; the equations may contain one, and only one, common factor, and may be both of
them satisfied, and yet the common factor may not vanish. The necessary condition
seems to be, that the one equation should be a factor of the other; in the case where
the assumed equation is of an order equal or superior to the matrix, then if this
equation contain as a factor the equation which is always satisfied by the matrix, the
assumed equation will be satisfied identically, and the condition is sufficient as well
as necessary: in the other case, where the assumed equation is of an order inferior
to that of the matrix, the condition is necessary, but it is not sufficient.
152]
A MEMOIR ON THE THEORY OF MATRICES.
487
30. The equation satisfied by the matrix may be of the form M^ = 1 ; the
matrix is in this ease said to be periodic of the ?ith order. The preceding conside
rations apply to the theory of periodic matrices; thus, for instance, suppose it is
required to find a matrix of the order 2, which is periodic of the second order. Writing
Jf=( a, 6 ),
c, d
we have
and the assumed equation is
Jlf»(a + d)if+a(i6c = 0,
:ilf «  1 = 0.
These equations will be identical if
a + d = 0, a(i  6c = — 1,
that is, these conditions being satisfied, the equation Jf * — 1 = required to be satisfied,
will be identical with the equation which is always satisfied, and will therefore itself
be satisfied. And in like manner the matrix M of the order 2 will satisfy the
condition Jf'— 1 = 0, or will be periodic of the third order, if only Jf'— 1 contains as
a factor
if "  (a + d) M'+ a(i  6c,
and so on.
31. But suppose it is required to find a matrix of the order 3,
M^{ a,
d,
6, c)
e. f
which shall be periodic of the second order. Writing for shortness
a — Jf,
h
,
M, f
^iM^AM^ + BM^C),
the matrix here satisfies
M*AM* + BMC = 0,
and, as before, the assumed equation is Jf ' — 1 = 0. Here, if we have 1+J5 = 0, A + C=^0,
the lefthand side will contain the £Ekctor (if'— 1), and the equation will take the form
(ilf* — 1) (if + C) = 0, and we should have then if' —1 = 0, provided M+C were not an
indeterminate matrix. But M+C denotes the matrix
(
a+C, b
)
d
9
e + C. f
h , i + C
488
A MEMOIR ON THE THEORY OF MATRICES.
[152
the determinant of which is C* + 4C*+J5(7+ C, which is equal to zero in virtue of
the equations 1+B = 0, 44C = 0, and we cannot, therefore, from the equation
(if»l)(if+C) = 0, deduce the equation if«l = 0. This is as it should be, for the
two conditions are not sufficient, in fact the equation
Jf« =
( a^ +bd +cgy ab + be + ch, ac + bf+ d )
da + ed +fg, db + ei^ ^fh, dc + ef¥fi
ga +hd + ig, gb + he + ih, gc + hf+ i*
= 1
gives nine equations, which are however satisfied by the following values, involving in
reality four arbitrary coefficients; viz. the value of the matrix is
(
 (7 f g) fiv'
a + /8 + 7
a + /8 + 7
^(0 + y)vfl'" ^(0+y)vfjr^ )
a + /8 + 7
 (7 + g) Xfi^
a + fi + y
o + i8 + 7
/9
a + 13 + y
 (0 + /3) I/X'
04/8 + 7
a + /8 + 7
so that there are in all five relations (and not only two) between the coefficients of
the matrix.
32. Instead of the equation M^ — 1=0, which belongs to a periodic matrix, it is
in many cases more convenient, and it is much the same thing to consider an
equation M^ — k = 0, where A; is a single quantity; The matrix may in this case be
said to be periodic to a factor pris.
33. Two matrices L, M are convertible when LM = ML. If the matrix M is given,
this equality affords a set of linear equations between the coefficients of L equal in
number to these coefficients, but these equations cannot be all independent, for it is
clear that if X be any rational and integral function of M (the coefficients being single
quantities), then L will be convertible with Jf ; or what is apparently (but only appa
rently) more general, if L be any algebraical function whatever of M (the coefficients
being always single quantities), then L will be convertible with M. But whatever the
form of the function is, it may be reduced to a rational and integral function of an
order equal to that of M, less unity, and we have thus the general expression for
the matrices convertible with a given matrix, viz. any such matrix is a rational and
integral function (the coefficients being single quantities) of the given matrix, the
order being that of the given matrix, less unity. In particular, the general form of
the matrix L convertible with a given matrix M of the order 2, is Z = aAf + /8, or
what is the same thing, the matrices
( a, 6 ), ( a\ V )
c, d
c', d'
will be convertible if a' — d' : 6' : c' = a — d : 6 : c.
152]
A MEMOIR ON THE THEORY OF MATRICES,
489
34. Two matrices Z, M are skew convertible when LM = — ML ; this is a relation
much less important than ordinary convertibility, for it is to be noticed that we cannot
in general find a matrix L skew convertible with a given matrix M. In fact, con
sidering M as given, the equality affords a set of linear equations between the coeffi
cients of L equal in number to these coefficients; and in this case the equations are
independent, and we may eliminate all the coefficients of Z, and we thus arrive at a
relation which must be satisfied by the coefficients of the given matrix M, Thus,
suppose the matrices
( a, 5 ), ( a\ V )
c,
c\ d!
are skew convertible, we have
(a, h){a\ 6' ) = ( oa' + fcc', oJ' + W ),
0, d \ d y dl
ca' + dc', cV + dd!
(a\ 5' )( a, 5) = (oa' + 6'c, a'5+6'rf),
c\ d! \ c, d
and the conditions of skew convertibility are
c'a + d'c, c'b + d'd
2oa' + 5c' + b'c = 0,
5' (a +rf)+6(a'+d') = 0,
c'{a +d) +c(a' + dO = 0»
2dd' + bc' + b'c =0.
Eliminating a\ b\ c\ d\ the relation between a, 5, c, d is
2a.
c ,
b , .
b ,
a +d,
b
c ,
•
a + d, c
•
c ,
b , 2d
= 0,
which is
(a + dy (adbc)^ 0.
Excluding from consideration the case orf — 6c = 0, which would imply that the matrix
was indeterminate, we have a + d = 0. The resulting system of conditions then is
a + d = 0, a' + d' = 0, aa' + 6c' + 5'c + dd' = 0,
the first two of which imply that the matrices are respectively periodic of the second
order to a factor pris,
35. It may be noticed that if the compound matrices LM and ML are similar,
they are either equal or else opposite ; that is, the matrices L, M are either convertible
or skew, convertible.
C. II. 62
490
A MEMOIR ON THE THEORY OF MATRICES.
[152
36. Two matrices such as
( a, 6 ), ( a, c ),
c, d
6, d
are said to be formed one fix)m the other by transposition, and this may be denoted
by the symbol tr. ; thus we may write
(a, c ) = tr. ( a, 6 ).
6, d \
c, d
The effect of two successive transpositions is of course to reproduce the original matrix.
37. It is easy to see that if M be any matrix, then
(tr. My = tr. (MP),
(tr. M)' = tr. ( Jf0.
and in particular,
38. If Z, if be any two matrices,
tr. (Zi/) = tr. il/. tr.Z,
and similarly for three or more matrices, L, M, N, &c.,
tr. (LMN) == tr. N. tr. M, tr. i, &c.
40. A matrix such as
(a, A, g ) '
K 6, /
which is not altered by transposition, is said to be symmetrical.
41. A matrix such as
( 0, V, m)
V, 0, X I
A^ X, !
which by transposition is changed into its opposite, is said to be skew symmetrical.
42. It is easy to see that any matrix whatever may be expressed as the sum of
a s}rmmetrical matrix, and a skew sjrmmetrical matrix ; thus the* form
hv, 6., / + X
flr + M, /X, c
which may obviously represent any matrix whatever of the order 3, is the sum of the
two matrices last before mentioned.
152]
A MEMOIB ON THE THEORY OF MATRICES.
491
43. The following formulae, although little more than examples of the composition
transposed matrices, may be noticed, viz.
(a, 6 $ a, c ) = ( a' + 6', ac{bd )
c, d
d, 5
ac^bd, c' + d'
which shows that a matrix compounded with the transposed matrix gives rise to a
symmetrical matrix. It does not however follow, nor is it the fact, that the matrix and
transposed matrix are convertible. And also
(a, c ^ a, 5 $ a, c ) = ( a» + 6cd + a(6» + c'), c» +a6d + c(a« + cP) )
b, d c, d \ \ b, d
which is a remarkably symmetrical form. It is needless to proceed further, since it
is clear that
(a, c ^ a, 6 $ «> ^ a, 6 )=(( a, c 1^ a, b ))*.
b, d \ c, d b, d c, d
b, d Cf d
44. In all that precedes, the matrix of the order 2 has frequently been con
sidered, but chiefly by way of illustration of the general theory ; but it is worth while to
develope more particularly the theory of such matrix. I call to mind the fundamental
properties which have been obtained, viz. it was shown that the matrix
satisfies the equation
and that the two matrices
M = ( a. 5 ),
1 c, a !
M*{a + d)M + adbc = 0,
(a, b). ( a\ V ),
c, d
will be convertible if
c', d'
a! — d' : V ', c' ^a^d : b : Ct
and that they will be skew convertible if
the first two of these equations being the conditions in order that the two matrices
may be respectively periodic of the second order to a factor jyrhs,
45. It may be noticed in passing, that if Z, Jlf are skew convertible matrices of
the order 2, and if these matrices are also such that i> = — 1, ilf' = — 1, then putting
JV=Z3f= — JfL, we obtain
Z«=l. if«=l, i<r« = i,
L^MN^^NM, M^^NL^^NL, N^LM^ML,
which is a system of relations precisely similar to that in the theory of quatemiona
62—2
492
A MEMOIR ON THE THEORY OF MATRICES.
[152
46. The integer powers of the matrix
Jf = ( a, 5 \
c, d
are obtained with great facility from the quadratic equation ; thus we have, attending
first to the positive powers,
Jtfa = (a 4 d)M(ad  be),
iV» = [(a + d)«  (orf  6c)] if  (a + d) (orf  6c),
&c.,
whence also the conditions in order that the matrix may be to a factor pris periodic of
the orders 2, 3, &c. are
a + d * = 0,
(a + dy  (ad  6c) = 0,
&c. ;
and for the negative powers we have
(od 6c) ¥' =  il/ + (a + d),
which is equivalent to the ordinary form
{ad''bc)M' = { . d, 6 );
c, a I
and the other negative powers of M can then be obtained by successive multiplications
with M\
47. The expression for the nth power is however most readily obtained by means
of a particular algorithm for matrices of the order 2.
Let A, 6, c, J, q he any quantities, and write for shortness iJ = — A' — 46c; suppose
also that h\ h\ c', tT, q' are any other quantities, such nevertheless that A' : 6' : c' = A : 6 : c,
and write in like manner iJ* = — A'* — 46V. Then observing that —r^ . —r . — are
respectively equal to f=^,> "7=57 » "7^* *^^ matrix
{ j( , h\ 2hJ
)
2cJ
'JR
.J (cot, + 4)
contains only the quantities J, q, which are not the same in both systems; and we
may therefore represent this matrix by (J, q\ and the corresponding matrix with
152]
A MEMOIR ON THE THEORY OF MATRICES.
493
h\ h\ c\ J\ q' by {J\ q'). The two matrices are at once seen to be convertible (the
assumed relations h' : h' : c' — h : h : c correspond in fact to the conditions,
a'^d! : b* : c' = a — d : b : c,
of convertibility for the ordinary form), and the compound matrix is found to be
\sin or sm a ^ ^ /
and in like manner the several convertible matrices {J, q), (J\ q'\ (J'\ q") &c. give
the compound matrix
Vsinasma smg ... ^ ^ i j
48. The convertible matrices may be given in the first instance in the ordinary^
form, or we may take these matrices to be
( a, 6 ), ( a\ V ), ( a", 6" ), &c.
c, d
c\ d'
c\ d"
where of course d — a : b : c = d* — a' : b' : cf = d" — a" : b'* : c** ^ &c. Here writing
h = d — a, and consequently R = — {d — o,y — 46c, and assuming also «/ = J VS and
cot a = — , — , and in like manner for the accented letters, the several matrices are
respectively
(i ^R. q) (4 >^, q'), (i ViP'. q"), &c..
and the compound matrix is
/ 8in( «y + q' + q" ^ ^^ ^g> ^ ^ + ,' + y"+ ...) .
49. When the several matrices are each of them equal to
(a, 5 ),
c, d
we have of course g = g' = g" . . . , R = R = Itf\,, , and we find
( a, 6 )
c, c2
=(s«^. "5)^
or substituting for the righthand side, the matrix represented by this notation, and
putting for greater simplicity
^p (i v:r)» = (J m L,oT L= ^l^i"? (i vs)».
494 A MEMOIR ON THE THEORY OF BfATRICE& [152
we find
( a. b )»=:(iZ(v^cotn9(da)), U )
\ c, d \ Lc , i Z (Vii cot wj + (d — a)) ;
where it will be remembered that
B = — (d — a)* — 46c and cot 9 = ._ ,
the last of which equations may be replaced by
. J — 1 . d + a + V^TS
cosg + v — l8ma= 7 ,
^ ^ 2Vad6c
The formula in £Ekct extends to negative or fractional values of the index n, and when
n is a fraction, we must, as usual, in order to exhibit the formula in its proper
generality, write q + 2nnr instead of q. In the particular case n » ^, it would be easjr
to show the identity of the value of the square root of the matrix with that before
obtained by a different process.
50. The matrix will be to a factor pris, periodic of the nth order if only sinitf^O,
that is, if 9 =  (m must be prime to n, for if it were not, the order of periodicity
would be not n itself, but a submultiple of n) ; but cos q = — . , and the condition
2 V od — be
is therefore
(d + o)» 4 (ad 6c) COS' "^ = 0,
or as this may also be written,
(P^a^^ 2ad cos h 46c cos* — = 0,
n n
a result which agrees with those before obtained for the particular values 2 and 3
of the index of periodicity.
51. I may remark that the last preceding investigations are intimately connected
QX \~ h
with the investigations of Babbage and others in relation to the function <f)X = ^ .
I conclude with some remarks upon rectangular matrices.
52. A matrix such as
( a, 6, c )
I a', 6', c I
where the number of columns exceeds the number of lines, is said to be a broad
matrix ; a matrix such as
( a, 6 )
where the number of lines exceeds the number of columns, is said to be a deep matrix.
152j
A MEMOIR ON THE THEORY OF MATRICES.
495
53. The matrix zero subsists in the present theory, but not the matrix unity.
Matrices may be added or subtracted when the number of the lines and the number
of the columns of the one matrix are respectively equal to the number of the lines
and the number of the columns of the other matrix, and under the like condition
any number of matrices may be added together. Two matrices may be equal or
opposite, the one to the other. A matrix may be multiplied by a single quantity,
giving rise to a matrix of the same form ; two matrices so related are similar to
each other.
54. The notion of composition applies to rectangular matrices, but it is necessary
that the number of lines in the second or nearer component matrix should be equal
to the number of columns in the first or further component matrix; the compound
matrix will then have as many lines as the first or further component matrix, and
as many columns as the second or nearer component matrix.
55. As examples of the composition of rectangular matrices, we have
{a.b,cji a\ b\ c\ d' ) = ( (a, 5, cJia\ e\ iO> («. b, c$h\f\ f) (a, 6, cl^c^ g\ k'\ (a, 6, c$d', K V) ),
d,ej
e\f\g\K
(d, e,/$< e\ i% (d, e,/$6', /, /) (d, e,/$c', fir', ^ (d, 6,/K. K V)
and
( a, d $ a', V, d, d') ) = ( (a. d$a'. e'). («. ^F. A («. ^<^> 9'\ («. ^d', h') ).
(6, e$a'. e'), (h, eji'./), (6. e$c'. (/'), {h. ejd', h!)
(c, /$a'. e'), (c. /$6', /), (c, /$c'. 9')> (c /K. A')
c./l
56. In the particular case where the lines and columns of the one component
matrix are respectively equal in number to the columns and lines of the other com
ponent matrix, the compound matrix is square, thus we have
( o, 6, c $ a', d' ) = ( (a. h, c$o', h', c'), (a, h, c^d', e', f) )
d, e, /
V, e'
(d. e. f^a', b', cO, (d, e, f^d', e', /')
and
( a', d' $ a, 6, c ) = ( (a', d'$a, d), (a', d'$6, e), (a', d'^c, f) ).
V, e'
c'.f
d, e,f
{V, ii \a, d). (6', e' $6, e), (6', c' $c, /)
(c', f\a, d), (c', f'\h, e). (c'. /'$c, /)
The two matrices in the case last considered admit of composition in the two different
orders of arrangement, but as the resulting square matrices are not of the same order,
the notion of the convertibility of two matrices does not apply even to the case in
question.
57. Since a rectangular matrix cannot be compounded with itself, the notions of
the inverse or reciprocal matrix and of the powers of the matrix and the whole resulting
theory of the functions of a matrix, do not apply to rectangular matrices.
496
A MEMOIR ON THE THEORY OF MATRICES.
[152
58. The notion of transposition and the symbol tr. apply to rectangular matrices,
the effect of a transposition being to convert a broad matrix into a deep one and
reciprocally. It may be noticed that the symbol tr. may be used for the purpose of
expressing the law of composition of square or rectangular matrices. Thus treating
(a, 6, c) as a rectangular matrix, or representing it by (a, 6, c), we have
tr. ( a\ b\ c' ) = ( a ),
and thence
( a, b, c ) tr. ( a\ h\ c' ) = (a, 6, cY a' )
= (a, 6, c\a\ h\ c'),
so that the symbol
(a, 6, c\a\ b\ c*)
would upon principle be replaced by
( a, 5, c ) tr. (a', 5', c' ):
I III
it is however more convenient to retain the symbol
(a, 6, c^a\ b\ c^).
Hence introducing' the symbol tr. only on the lefthand sides, we have
^ a, 6, c ) tr. ( a, b\ cM = ( (a, 6, cja\ b\ c'), (a, 6. c$d', e\ f) ),
d, ^. / I i d\ ^, /' I I (d. e, fla\ b\ c% (d, e, fJid\ ^. /O 
or to take an example involving square matrices.
( a, 5 ) tr. ( a', 6' ) = ( (a, 6$a', 5'), (a, 6$d', e') ) ;
I d, e I I d', e' I I (d, c$a', 6'), (d, e^d\ e')
it thus appears that in the composition of matrices (square or rectangular), when the
second or nearer component matrix is expressed as a matrix preceded by the symbol
tr., any line of the compound matrix is obtained by compounding the corresponding
line of the first or further component matrix successively with the several lines of the
matrix which preceded by tr. gives the second or nearer component matrix. It is clear
that the terms 'symmetrical* and *skew symmetrical* do not apply to rectangular
matrices.
153]
497
153.
A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION
OF A BIPARTITE QUADRIC FUNCTION.
[From the Philosophical Transactions of the Royal Society of London, vol. cxLvm. for the
year 1858, pp. 39 — 4!6. Received December 10, 1857, — Read January 14, 1858.]
The question of the automorphic linear transformation of the function a^ •\ y^ •¥ s^,
that is the transformation by linear substitutions, of this function into a function
a?/ + y/ + V of the same form, is in eflfect solved by some formulae of Euler's for the
transformation of coordinates, and it was by these formulae that I was led to the
solution in the case of the sum of n squares, given in my paper "Sur quelques pro
pri^tds des determinants gauches"(0 ^ solution grounded upon an d priori investiga
tion and for the case of any quadric function of n variables, was first obtained by
M. Hermite in the memoir "Remarques sur une M^moire de M. Cayley relatif aux
determinants gauches"(') This solution is in my Memoir "Sur la transformation d'une
function quadratique en ellememe par des substitutions lin6aires"('), presented under a
somewhat different form involving the notation of matrices. I have since found that
there is a like transformation of a bipartite quadric function, that is a lineolinear
function of two distinct sets, each of the same number of variables, and the develop
ment of the transformation is the subject of the present memoir.
1. For convenience, the number of variables is in the analytical formulae taken
to be 3, but it will be at once obvious that the formulae apply to any number of
variables whatever. Consider the bipartite quadric
( a , 5 , c $a?, y, ^][x, y, z),
a' , 6' , c'
a y , c
1 CreUe, X. xxxn. (1846) pp. 119—123, [62].
' Cambridge and Dublin Mathematical Journal^ t. ix. (1854) pp. 63 — 67.
» CrelU, t. L. (1866) pp. 288—299, [186].
c. II. 63
498
A MEMOm ON THE AUTOMORPHIC LINEAR TRANSFORMATION
[153
which stands for
(ax + 6y hcz )x
+ (a'x + Vy +c'z)y
+ (a"a? + V'y + c''z) z,
and in which {x^ y, z) are said to be the nearer variables, and (z, y, z) the further
variables of the bipartite.
2. It is clear that we have
( a , 6 , c $a?, y, ^$x, y, z) = ( a, a\ a" $x, y, z$a?, y, z)
«'' j»'' •"
a , , c
6, y, 6"
c, c , c
//
and the new form on the rightband side of the equation may also be written
(tr. ( a , 6 , c ) $x, y, z^x, y, z),
^" !»" V
a , , c
that is, the two sets of variables may be interchanged, provided that the matrix is
transposed
3. Each set of variables may be linearly transformed: suppose that the substitu
tions are
(«, y, 8^) = ( i , wi , n $a?^, y,, z,)
and
(X, y. z) = ( 1 . 1' . 1" $^,. y,. O
//
m, m, m
n , n , n
Then first substituting for {x, y, z) their values in terms of (a?^, y^, z^\ the bipartite
becomes
( ( a , 6 , c 5^ Z , m , n ) $a?„ y„ ^,$x, y, z);
a", 5", c"
represent for a moment this expression by
( A , B , C $a?,, y,, z,^x, y, z),
153]
OF A BIPARTITE QUADRIC FUNCTION.
499
then substituting for (x, y, z) their values in terms of (x^, y^, z^), it is easy to see
that the expression becomes
( ( 1 , m , n $ il , 5 , ) $/c„ y,, zjx„ y„ z,),
r, m'', n'
A\ R, cr
and reestablishing the value of the auxiliary matrix, we obtain, as the final result of
the substitutions,
( a , 5 , c $a?, y, «$x, y, z) = (( 1 , m , n ISa ,h , c \l , m , n ) $a?,, y,, ^^$x,, y,, zj,
_// L/' >."
r, m', n'
r, m", n'
a', 6', c'
_// L// //
a , , c
that is, the matrix of the transformed bipartite is obtained by compounding in order,
first or furthest the transposed matrix of substitution of the further variables, next the
matrix of the bipartite, and last or nearest the matrix of substitution of the nearer
variablea
4. Suppose now that it is required to find the automorphic linear transformation
of the bipartite
( a , 6 , c\ X, y, z^, y, z),
a' , V y c'
^" u* V
a , , c
or as it will henceforward for shortness be written,
(n$a?, y, ^$x, y, z);
this may be eflfected by a method precisely similar to that employed by M. Hermite
for an ordinary quadric. For this purpose write
a? + a?^ = 2f, y + y, = 2i7, z + ^, = 2f,
x + x, = 2B, y + y, = 2H, z + z^ = 2Z,
or, as these equations may be represented,
(a?+a:„ y)ry,, z + f,)= 2(f , 17 , f ),
(x + x„ y + y,, z + z^)=2(H, H, Z);
then we ought to have
(n$2fa?, 217 y, 2f.^][2ax, 2Hy, 2Z  z) = (n][a;, y, erjx, y, z).
5. The lefthand side is
4(ft$f, ^, ?$H, H, Z)2(n$a?, y, ^$B, H, Z)2(n$f 17, ?$x, y, z) + n$^, y, ^$x, y, z),
and the equation becomes
2(n$f 1,. r$B. H. Z)(fl$a:, y, r$S. H. Z)(ft$f. ,,. r$x, y. z)=:0,
63—2
500 A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION [153
or as it may be written,
("$f , V. ?$B, H, Z) . (a^x, y, ^$B, H, Z)
. y. z))
X, Hy. Zz){ '
X. vy. ?«$a. H, Z)]^
X, Hy. Zz$f. ,. ?)) '
+ (n$f. V. ?$B. H, z)  (n$f, ,, r$x
or acfain,
^' (n$f.. ,y, ?.$B. H. Z)
+ («$?. I?. ?$B
or what is the same thing,
(ft$f^, vy. ?^$H, H, Z)
+ (tr. njB
and it is easy to see that the equation will be satisfied by writing
(tr.n$Hx, Hy, Zz) = (tr.T$B, H, Z).
where T is any arbitrary matrix. In fact we have then
( n$f X, V y, K ^$B. H, Z)= ( T$f , , , r$B, H. Z),
(tr.n$Sx. Hy, Zz$f . i, , r) = (tr.T$H, H, Z$f , ,,. f)
( T$f , ^ , ?$B. H, Z),
and the sum of the two terms consequently vanishe&
6. The equation
gives
(nT$f, ,, K)=mj'.y.').
and we then have
In fact the two equations give
or what is the same thing,
which is the equation assumed as the definition of (^, 17, ^); and conversely, this
equation, combined with either of the two equations, gives the other of them.
7. We have consequently
(^, y, z) = (n^ (ft . T)$f , 17. ?),
(f, 17, ?) = ((" + T)^"$^., y.»0»
and thence
(X. y, *)=(n>(nT)(n+T)>n$^„ y„ o.
153] OF A BIPARTITE QUADRIC FUNCTION. 501
8. But in like maimer the equation
(tr.n$Hx, Hy, Z  z) =  (tr. T$B, H, Z)
gives
(tr. fl + T$E, H, Z) = (tr. n$x, y, z),
and we then obtain
(tr.fl^TJB, H, Z) = (tr.ft$x„ y,, z,).
9. In tact these equations give
(tr.2n$B, H, Z) = (tr.fl$x + x„ y + y„ z + z^),
or
2(a. H, Z) = (x + x,, y + y„ z + z,);
and conversely, this equation, combined with either of the two equations, gives the other
of them. We have then
(x, y , z) = ((tr.ft)ntr.fl + T$H, H, Z),
(B, H, Z) = ((tr.nT)^ tr. n$x,, y,, z>
and thence
(X, y, z) = ((tr. ft)» (tr. flTT)(tr. flT)Hr. ftjx,, y„ z,).
10. Hence, recapitulating, we have the following theorem for the automorphic linear
transformation of the bipartite
(n$a?, y, z^x, y, z),
viz. T being an arbitrary matrix, if
(x, y, ^)=:(flnnT)(fl + T)^n$a;,, y„ O,
(x, y, z) = ((tr.n)Mtr. n + T)(tr. flT)»tr. n$x„ y„ z,),
then
(ft$a?, y, ^$x, y, z) = (n$a?,, y„ s,$x„ y„ z),
which is the theorem in question.
11. I have thought it worth while to preserve the foregoing investigation, but
the most simple demonstration is the verification d posteriori by the actual substitution
of the transformed values of (x, y, z), (x, y, z). To effect this, recollecting that in general
tr. (il~*) = (tr. 4)"* and tr. ABCD ^ tr, D. tr. C, tr. B. tr. A, the transposed matrix of
substitution for the further variables is
ft (ft  T)Hn + T) n^
and compounding this with the matrix ft of the bipartite, and the matrix
ftMftT)(ft+T)^ft
502 A MEMOIR ON THE AUTOMOBPHIC LINEAR TRANSFORMATION [l53
of substitution for the nearer variables, the theorem will be verified if the result k
equal to the matrix fl of the bipartite ; that is, we ought to have
or what is the same thing,
11(1:1  T)Hft + T) nHfl  T)(ft + T)^fl = n ;
this is successively reducible to
(n h T)ft>(n  T) = (ft  T)ft'(ft + T),
ft^ft + T)ft»(ft  T) = ft^ft  T)ft'(ft + T),
(1 + ft'T)(l  ft^T) =(l.ft*T)(l + fl^T),
which is a mere identity, and the theorem is thus shown to be true.
12. It is to be observed that, in the general theorem, the transformations or matrices
of substitution for the two sets of variables respectively are not identical, but it may
be required that this shall be so. Consider first the case where the matrix ft is
sjnumetrical, the necessary condition is that the matrix T shall be skew symmetrical ;
in fact we have then
tr. ft = ft, tr. T=T,
and the transformations become
(x, y, ^) = (ftHftT)(ft+T)»ft$a:,, y„ z,\
(X, y, z) = (ft^(ft  T)(ft + T)^ft$x„ y„ z,),
which are identical. We may in this case suppose that the two sets of variables
become equal, and we have then the theorem for the automorphic linear transformation
of the ordinary quadric
(ft$a?, y, zy,
viz. T being a skew sjnumetrical matrix, if
(x, y, ^) = (ft^n  T)(ft + T)^ft$a?„ y„ z,),
then
(ft$a?, y, zy = (ft$a;„ y„ z,y,
13. Next, if the matrix ft be skew symmetrical, the condition is that the matrix
T shall be sjnumetrical ; we have in this case tr. ft = — ft, tr. T = T, and the four factors
in the matrix of substitution for (x, y, z) are respectively — ft~*, — (ft — T), — (ft + T)*
and —ft, and such matrix of substitution becomes therefore, as before, identical with
that for (a?, y, z)\ we have therefore the following theorem for the automorphic linear
transformation of a skew symmetrical bipartite
(ft$a?, y, «$x, y, z),
153] OF A BIPARTITE QUADRIC FUNCTION. 503
when the transformations for the two sets of variables are identical, viz. T being any
symmetrical matrix, if
{X. y. ^) = (fi'(nT)(n + T)'n$«,. y„ z).
(X, y, z) = (n>(fiT)(n + T)>ft$x„ y„ z,).
then
(ft$a?, y, £$x, y, z)=(ft$aj,, y„ z,$x,, y„ z,).
14. Lastly, in the general case where the matrix H is anything whatever, the
condition is
fl^T =  (tr. n)* tr. T
for assuming this equation, then first
n^n  T) = (tr. n)Htr. fn t),
and in like manner
Qr\a + T) = (tr. n)*(tr. n  T).
But we have
1 = (tr. n)i(tr. n  T) (tr. n  T)» tr. n,
and therefore, secondly,
(ft + T)* ft = (tr. ft  T)* tr. ft ;
and thence
ft^n  T)(ft + T)»ft = (tr. ft)» (tr. ft + T) (tr. ft  T)Hr. ft,
or the two transformations are identical.
15. To further develope this result, let ft"* be expressed as the sum of a
symmetrical matrix Q^ and a skew symmetrical matrix Q^, and let T be expressed in
like manner as the sum of a symmetrical matrix T^ and a skew symmetrical matrix
T^. We have then
(tr.ft)* = tr.(ft0 = eoe.,
T =T, + T„
tr. T =ToT„
and the condition, ft"* T = — (tr. ft)* tr. T, becomes
that is,
<2.T, + Q,T, = 0,
and we have
504
A MEMOIR ON THE AUTOMORPHIC LIXEAB TRANSFORBCAHON
[153
or as we may, write it.
and thence
To =  (i{n» + tr. ft»})»(J{ft^  tr. n»})T,.
T =  (i{fl^ + tr. ft»})Ki{«*  tr. fl»})T, + T,.
where T^ is an arbitrary skew sjnmmetrical matrix.
16. This includes the beforementioned special cases; first, if fi is symmetrical,
then we have simply T^T^, an arbitrary skew symmetrical matrix, which is right
Next, if fl is skew symmetrical, then T = — 0*n~*T^ + T^, which can only be finite
for T^ = 0, that is, we have T = O*n"*0, and (the first part of T being always
sjnumetrical) this represents an arbitrary s}'mmetrical matrix. The mode in which this
happens will be best seen by an exampla Suppose
fli = ( A , jy+i'), tr. ft»=( A , H^v\
and write
then we have
T, = ( 0, 6),
0,0
r={A, ff)>( 0. !.)( 0, e)+( 0. 0)
H, b\ i;, o! I^. 0' 1^,
v0
±jn{B, H) + ( 0, 0)
ABH
H, A \0,
_ ( vB0
ABH*
vH0 ^^) + (
pH0
ABH
,^.
ABH'
vA0
ABH*
0, 0)
0,
When n is skew symmetrical. A, B, H vanish; but since their ratios remain arbitrary,
we may write kA, kB, kH for A, B, H, and assume ultimately « = 0. Writing k0
in the place of 0, and then putting k= 0, the matrix becomes
( vB0
vB0 )
ABH*' ABH*
vH0
vA0
ABH*' ABH*
which, inasmuch aa A .0, B : 0, and C : remain arbitrary, represents, as it should do,
an arbitrary symmetrical matrix.
153] OF A BIPARTITE QUADRIC FUNCTION. 505
17. Hence, finally, we have the foUoMring Theorem for the automorphic linear
transformation of the bipartite qnadric,
(ft$a?, y, z^x, y, z),
when the two transformations are identical, viz. if T^ be a skew symmetrical matrix,
and if
T =  (i{n^ + tr. fl^})(J{n^  tr. fl^})T, + T, ;
then if
(X. y, «) = (fl>(nT)(n + T)>n$ar„ y,. z),
(X, y, z) = (fl> (n  T) (ft + T)> n$x„ y,. z,);
we have
(n$a?, y, ^$x, y, z) = (n$a?„ y„ z,^x^, y„ z,);
and in particular.
If n is a sjnnmetrical matrix, then T is an arbitrary skew sjrmmetrical matrix ;
If n is a skew symmetrical matrix, then T is an arbitrary symmetrical matrix.
c. n.
64
506
[154
154.
SUPPLEMENTARY RESEARCHES ON THE PARTITION OF
NUMBERS.
[From the PhilosophiccU Transactions of the Royal Society of London^ vol. cxlviil for
the year 1868, pp. 47—52. Received March 19,— Read June 18, 1857.]
The general formula given at the conclusion of my memoir, "Researches on the
Partition of Numbers "(0> ^ somewhat diflferent ftom the corresponding formula of
Professor Sylvester', and leads more directly to the actual expression for the number of
partitions, in the form made use of in my memoir; to complete my former researches,
I propose to explain the mode of obtaining from the formula the expression for the
number of partitions.
The formula referred to is as follows, viz. if ^ be a rational fraction, the denomi
fx
nator of which is made up of &ctors (the same or different) of the form 1—^, and
if a is a divisor of one or more of the indices m, and k is the number of indices of
which it is a divisor, then
where
1 Philotophical Transactions, torn, oxlvi. (1S56) p. 127, [140].
' Professor Sylvester's researches are published in the Quarterly Mathematical Journal, torn. i. [1857,
pp. 141 — 152]; there are some numerioal errors in his value of P (1, 2, 8, 4, 5, 6) q.
154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OP NUMBERS. 507
in which formula [1 — of*] denotes the irreducible &ctor of 1 — o^, that is, the fS^^tor
which equated to zero gives the prime roots, and /d is a root of the equation
[1 — a?*] = ; the summation of course extends to all the roots of the equation. The
index 8 extends from « = 1 to 8=^k; and we have then the portion of the fraction
depending on the denominator [1— «"]. In the partition of numbers, we have ^ = 1,
and the formula becomes therefore
{
71 ='... + rT7^i^('^'y~'S^^^■^
/«j[,_«.j n(«i)^ px
where
Uis'l)^ " [l^a^y
We may write
vp = coeff.  in i^^ ^. .. .
^^ t Ape')
/a? = n (1  ««),
where m has a given series of values the same or diflferent. The indices not divisible
by a may be represented by m, the other indices by ap, we have then
where the number of indices ap is equal to k. Hence
f(pe^) = n (1  p^'er^) H (1  p^er^p^y,
or since /> is a root of [! — «:*] = 0, and therefore />* = 1, we have
fipe^) = n (1  p^e^) n (1  er^J^)\
and it may be remarked that if n = i/ (mod. a), where v<a^ then instead of p** we
may write p", a change which may be made at once, or at the end of the process of
development.
We have consequently to find
>^ = ^^^i^^^'n(i.pneon(iei^)
The development of a fS^^tor ^ ^^ is at once deduced from that of . __ . , and is
a series of positive powers of t The development of a factor  _^^ap< ^^ deduced from
1 . 1 *
that of ;j — nj, and contains a term involving . Hence we have
n (1 p'e"*) n (1  e^) =""** ««'^^*"F5'^'*?'^'*'''*'^'
and thence
XP = P^*
64—2
508 SUPPLEMENTABT BESEABCHES ON THE PARTITION OF NUMBEBa [154
The actual development, when k is small (for instance k^\ or £ = 2), is most readily
obtained by developing each &ctor separately and taking the product. To do this we
have
where by a general theorem for the expansion of any function of e*, the coefficient
of ^^ is
=ty 1 (K
n/ica+A)"^
n/ vic • (ic)» — • (icy+^
<where as usual A(V = l>'0/ A«0/ = 2/2 . l^ + CK, &c.) and
1 111. 1^. l«p
l6« e"^2 12 720 30240
where, except the constant term, the series contains odd powers only and the coef
l—y^^Bf .111
ficient of t^'^ is n9/' ^ ^*' ^^' ^^'" ^^^^^^^8 *^® series ^, ^txi tS" ^^ Bernoulli's
numbers.
But when k is larger, it is convenient to obtain the development of the fraction
firom that of the logarithm, the logarithm of the fraction being equal to the sum of
the logarithms of the simple factors, and these being found by means of the formulae
The fraction is thus expressed in the form
n(i.p«)n(ap)f*
and by developing the exponential we obtain, as before, the series commencing with
Besuming now the formula
XP = P^*>
which gives p^ as a function of p, we have
Ox
[1  a^] p  a? '
154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 509
but this equation gives
and we have
[laj«] = (a?p)(a?p««)...(a?.p«a),
if 1, Oj, ... a. are the integers less than a and prime to it (a is of course the degree
of [1  ic*]). Hence
and therefore
or putting for yp its value
^p =  p« n (1  p«'0 A^,
where a is the degree of [I— a^] and o^ denotes in succession the integers (exclusive
of unity) less than a and prime to it. The function on the right hand, by means of
the equation [I— p*] = 0, may be reduced to an integral function of p of the degree
a — 1, and then by simply changing p into x we have the required function 0x, The
0x
fraction ^i — I5T ^^° then by multiplication of the terms by the proper factor be
reduced to a fraction with the denominator 1 — a;*, and the coefficients of the numerator
of this fraction are the coefficients of the corresponding prime circulator ( ) per a^.
Thus, let it be required to find the terms depending on the denominator [1 — a:*] in
(la?)(l.a:«)(la;»)(la?*)(la?»)(la;«)'
these are
where
p—x p—x
and
fipe') (1  per") (1  p»6^) (1  p'ir^) (1  p»r^) (1  O (1  e^)
510 SUFPLEMEMTABT RESEARCHES ON THE PARTITIOK OF NUMBEBS. [15^
where it is easy to see that
1 1
A^ =
18ilp)(lp')(lp^)(lp»)'
A 1 fl 1 f p ip' 4,p* ^p^ \]
^'~(lp)(lp')(lf^){lp')\* 18\lp^lf^'^lp*^lp^)]'
and we have
e^ = p*(lp)A^,
^,p = p»(lp)4_,.
But [1 — p*] = 1 + p + p* = 0. Hence p* = 1, and therefore
(1 p)(l p'Xl p*)(l p')(l p)«(l p')'=9.
Hence
<'sP = i^p'(lp) = 4(lp')=l^(2+p),
whence
and the partial fraction is
1 2hx
1621+a? + a'»'
which is
""162 1a^ '
and gives rise to the prime circulator t^ (2, —1. — l)pcr 3,.
The reduction 0ip is somewhat less simple; we have
= ^(lp')(5110p14p0
= ^ (61 + 4p  65p') ;
154] SUPPLEMENTARY BESEABCHES ON THE PARTITION OF NUMBERS. 511
hence finally
^^P = m^''^ '^ ^^f'^' ^i« = 3^(42 + 23a:);
and the partial fraction ia
which is
J_ . 42 + 23a!
S2* '1+x + a/"
1  4219a;23g'
324 '^^ r^
and gives rise to the prime circulator ssi. ? (42, — 19, — 23) per 3
324
The part depending on the denominator 1 — a; is
^' +1^0.^^ + A (^e.)* ^
lx 1 "la; ■ 1.2
where
lx
1
...+
1.2.3.4.5
(xd,y
A.
lx'
We have here
(1  O (1  e*) (1  «") (1  «^) (1  e«) (1  e«)
A i . A ^ ,1 1 P
= il. t; + d_,  ... + j4_,  +&C.
logi3^ = log« + 2<24^ + 2880^'^'
and thence the fraction is
, 21, 91„.4S5
720 f
which is equal to
720 «•
21 , . 441 ^ . 3087 ^ . 64827 ^ . 1361367 ^ ,
t + gt + ^g «• + 128" ^ + 1280 "''•••
!0 «• /, 21
/, 91^ 8281^ \
11 7 1 77 1 245 1 . 43981 1 . 199677 1
720 f*^ 480 «• 1080 «« 1152 f 103680 1? 345600 <
and consequently the partial fractions are
(ae.)» , + ^^^5a (^'Y ^— + cJ5a («®«)' 1 1 + ooAi (a^*)*
86400
1  a; ■ 11520
lx 6480
1a; ' 2304
103680
from which the noncirculating part is at once obtained.
. 43981 ,_^ , 1 ^
+ ,,>»^»^ (aW») :; +
lx
199577 1
1  a;^ 345600 la;'
512 SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. [154
The complete expression for the number of partitions is P (1, 2, 3, 4, 5, 6) q =
^QgggQQ (IV + 63(V + 12303» + 1102503» + 4398109 + 698731)
+ ^ (69' + 126g + 581)(l. 1) per 2,
+ Jg2? (2,  1,  1) per 3,
+ 3^4 (42, 19. 23) per 3,
+ 32 (1, 1,  1,  1) per 4,
+ ^ (2, 1, 0, 1.2) per 5,
+ ^ ..(2. 1,1, 2, 1,1) per 6^
155]
518
155.
A FOURTH MEMOIR UPON QUANTICS.
[From the Philosophical Transactions of the Royal Society of London, vol. cxlviil fir
the year 1858, pp. 415—427. Received February 11,— Read March 18, 1858.]
The object of the present memoir is the fiirther development of the theory of binary
quantics ; it should therefore have preceded so much of my third memoir, t. 147 (1857),
p. 627, [144], as relates to ternary quadrics and cubics. The paragraphs are numbered
continuously with those of the former memoira The* first three paragraphs, No& 62 to 64,
relate to quantics of the general form (♦$a?, y, ...)^i and they are intended to complete
the series of definitions and explanations given in Nos. 54 to 61 of my third memoir;
Nos. 68 to 71, although introduced in reference to binary quantics, relate or may be
considered as relating to quantics of the like general form. But with these exceptions
the memoir relates to binary quantics of any order whatever: viz. Nos. 65 to 80 relate
to the covariants and invariants of the degrees 2, 3 and 4; Nos. 81 and 82 (which are
introduced somewhat parenthetically) contain the explanation of a process for the
calculation of the invariant called the Discriminant; Nos. 83 to 85 contain the definitions
of the Catalecticant, the Lambdaic and the Canonisant, which are functions occurring in
Professor Sylvester's theory of the reduction of a binary quantic to its canonical form ;
and Nos. 86 to 91 contain the definitions of certain covariants or other derivatives
connected with Bezout's abbreviated method of elimination, due for the most part to
Professor Sylvester, and which are called Bezoutiants, Cobezoutiants, &a I have not in
the present memoir in any wise considered the theories to which the catalecticant, &c.
and the lastmentioned other covariants and derivatives relate; the design is to point
out and precisely define the different covariants or other derivatives which have hitherto
presented themselves in theories relating to binary quantics, and so to complete, as far
as may be, the explanation of the terminology of this part of the subject.
62. If we consider a quantic
(a, 6, ...$a?, y, ...)•"
c. II. 65
514 A FOURTH MEMOIR UPON QU ANTICS. [l55
and an adjoint linear form, the operative quantic
(a, 6, ...$3, 9,,...)'*,
or more generally the operative quantic obtained by replacing in any covariant of the
given quantic the facients (a?, y, ...) by the symbols of differentiation (9^. 9,,...) (which
operative quantic is, so to speak, a contravariant operator), may be termed the Pro
vector ; and the Provector operating upon any contravariant gives rise to a contra
variant, which may of course be an invariant. Any such contravariant, or rather such
contravariant considered as so generated, may be termed a Provectant; and in like
manner the operative quantic obtained by replacing in any contravariant of the given
quantic the facients (f , 17, ...) by the sjrmbols of differentiation (9jb, 9y, ...) (which operative
quantic is a covariant operator), is termed the Cantraprovector ; and the contraprovector
operating upon any covariant gives rise to a covariant, which may of course be an
invariant. Any such covariant, or rather such covariant considered as so generated,
may be termed a Contraprovectant
In the case of a binary quantic,
(a, 6, ...][a?, y)*",
the two theorems coalesce together, and we may say that the operative quantic
(a, 6, ...$9y,9«)"*,
or more generally the operative quantic obtained by replacing in any covariant of the
given quantic the facients (a?, y) by the symbols of differentiation (9y, — 9,) (which is
in this case a covariant operator), may be termed the Provector. And the Provector
operating on any covariant gives a covariant (which as before may be an invariant),
and which considered as so generated may be termed the Provectant.
63. But there is another allied theory. If in the quantic itself or in any covariant
we replace the facients (x, y, ...) by the first derived functions (9fP, 9^, ...) of any con
travariant P of the quantic, we have a new function which will be a contravariant of
the quantic. In particular, if in the quantic itself we replace the &cient8 (a?, y, ...) by
the first derived functions (9fP, 9^,...) of the Reciprocant, then the result will contain
as a &ctor the Reciprocant, and the other factor will be also a contravariant. And
similarly, if in any contravariant we replace the facients (f, 17,...) by the first derived
functions (9«TF, dyW,..,) of any covariant W (which may be the quantic itself) of the
quantic U, we have a new function which will be a covariant of the quantia And in
particular if in the Reciprocant we replace the facients (f, 17, ...) by the first derived
functions (dxU, dyU, ...) of the quantic, the result will contain IT as a fisu;tor, and the
other factor will be also a covariant. In the case of a binary quantic (a» 6, ...$«, y)*
the two theorems coalesce and we have the following theorem, viz. if in the quantic
U or in any covariant the facients (x, y) are replaced by the first derived functions
(^yWy — 9«TF) of a covariant TT, the result will be a covariant ; and if in the quantic
155] A FOURTH MEMOIR UPON QU ANTICS. 515
U the facients (a?, y) are replaced by the first derived functions (9yJ7, —dxU). of the
quantic, the residt will contain ^ as a factor^ and the other fiEU)tor will be also a
covariant.
Without defining more precisely, we may say that the function obtained by replacing
as above the facients of a covariant or contravariant by the first derived functions of a
contravariant or covariant is a Transmutant of the firstmentioned covariant or contra
variant.
64. Imagine any two quantics of the same order, for instance, the two quantics
F==(a', 6',...$a?, y ...)'",
then any quantic such as \UhfiV may be termed an Intermediate of the two quantics;
and a covariant of \U+fiV, if in such covariant we treat \, /i as facients, will be a
quantic of the form
where the coeflScients (A, B, ... R, A") will be covariants of the quantics U, F, viz. A
will be a covariant of the quantic U alone ; J." will be the same covariant of the quantic
V alone, and the other coefficients (which in reference to A, A^ may be termed the
Connectives) will be covariants of the two quantics; and any coefficient may be obtained
from the one which precedes it by operating on such preceding coefficient with the
combinantive operator
a'3a+6'96 + ...,
or fiom the one which succeeds it by operating on such succeeding coefficient with the
combinantive operator
ad a' + idb' + • • . >
the result being divided by a numerical coefficient which is greater by unity than
the index of fi or (as the case may be) \ in the term corresponding to the coefficient
operated upon. It may be added, that any invariant in regard to the fiaicients (X, fi)
of the quantic
(A, B, ... R, A'^\ fiY
is not only a covariant, but it is also a combinant of the two quantics 17, F.
As an example, suppose the quantics IT, F are the quadrics
(a, 6, c\x, yy and (a\ h\ c'\x, yf,
then the quadrinvariant of
XlT+ZiF is (Xa + fta'XXc + /ic')  (X6 + /iftO*.
which is equal to
{ac}?, ac'2hV + ca\ a'c'6'«5x, /*)»,
and oc' — 266' + ca' is the connective of the two discriminants oc — 6* and aV — 6'*.
65—2
516 A FOURTH MEMOIB UPON QUANTICS. [l55
65. The law of reciprocity for the number of the invariants of a binary quantic\
leads at once to the theorems in regard to the number of the quadrinvariantSy cabin
variants and quartinvariants of a binary qmmtic of a given degree, first obtained by
the method in the second part of my original memoir*. Thus a quadric has only a
single invariant, which is of the degree 2 ; hence, by the law of reciprocity, the number
of quadrinvariants of a quantic of the order m is equal to the number of ways in which
m can be made up with the part 2, which is of course unity or zero, according as
m is even or odd. And we conclude that
The quadrinvariant exists only for quantics of an even order, and for each such
quantic there is one, and only one, quadrinvariant.
66. Again, a cubic has only one invariant, which is of the degree 4, and the
number of cubinvariants of a quantic of the degree m is equal to the number of
ways in which m can be made up with the part 4. Hence
A cubinvariant only exists for quantics of an evenly even order, and for each
such quantic there is one, and only one, cubinvariant.
67. But a quartic has two invariants, which are of the degrees 2 and 3 respectively,
and the number of quartinvariants of a quantic of the degree m is equal to the number
of ways in which m can be made up with the parts 2 and 3. When m is even,
there is of course a quartinvariant which is the square of the quadrinvariant, and which,
if we attend only to the irreducible quartinvariants, must be excluded from consideration.
The preceding number must therefore, when m is even, be diminished by unity. The
result is easily found to be
Quartinvariants exist for a quantic of any order, even or odd, whatever, the quadric
and the quartic alone excepted; and according as the order of the quantic is
6(7, 6(7 + 1, 6(7 + 2, 6^ + 3, 6(7 + 4, 6(7 + 5,
the number of quartinvariants is
g, 9 f 9 » 5^ + 1, 9 > 5^+1
In particular, for the orders
2, 3, 4, 5; 6, 7, 8, 9, 10, 11; 12, &a,
the numbers are
0, 1, 0, 1; 1, 1, 1, 2, 1, 2; 2, &c
Thus the ninthic is the lowest quantic which has more them one quartinvariant.
68. But the whole theory of the invariants or covariants of the degrees 2, 3, 4 is
most easily treated by the method above alluded to, contained in the second part of my
original memoir; and indeed the method appears to be the appropriate one for the
I Introduistoiy Memoir, [189], No. 20. * Ibid. No8. 1017.
1
i
155] A FOUKTH MEMOIR UPON QUANTIC8. 517
treatment of the theory of the invariants or covariants of any given degree whatever,
although the application of it becomes difficult when the degree exceeds 4. I remark,
in regard to this method, that it leads naturally, and in the first instance, to a special
class. of the covariants of a system of quantics, viz. these covariants are linear functions
of the derived functions of any quantic of the system. (It is hardly necessary to remark
that the derived functions referred to are the derived functions of any order of the
quantic with regard to the facients.) Such covariants may be termed tantipartite
covariants; but when there are only two quantics, I use in general the term lineolinear.
The tantipartite covariants, while the system remains general, are a special class of
covariants, but by particularizing the system we obtain all the covariants of the par
ticularized system. The ordinary case is when all the quantics of the system reduce
themselves to one and the same quantic, and the method then gives all the covariants
of such single quantic. And while the order of the quantic remains indefinite, the
method gives covariants (not invariants); but by particularizing the order of the quantic
in such manner that the derived functions become simply the coefficients of the quantic,
the covariants become invariants: the like applies of course to a sjrstem of two or more
quantics.
69. To take the simplest example, in seeking for the covariants of a single quantic
Uj we in fact have to consider two quantics U, V, An expression such as 12 IT 7" is a
lineolinear covariant of the two quantics; its developed expression is
which is the Jacobian. In the particular case of two linear functions (a, 6][a?, y) and
(of, V^x, y)y the lineolinear covariant becomes the lineolinear invariant ab' — a% which
is the Jacobian of the two linear fiinctiona
In the example we cannot descend from the two quantics U,VU> the single quantic
U (for putting F= 17 the covariant vanishes); but this is merely accidental, as appears
by considering a diflferent lineolinear covariant 12* [/"F", the developed expression of
which is
In the particular case of two quadrics (a, 6, c$x, yf, (a\ b', (/'^x, y)\ the lineolinear
covariant becomes the lineolinear invariant
ac'  266' + ca'.
If we have V—U, then the lineolinear covariant gives the quadricovariant
d»^U.dy^U^(dJdyUy
of the single quantic U (such quadricovariant is in &ct the Hessian) ; and if in the last
mentioned formula we put for U the quadric (a, 6, c][, x, yf, or what is the same thing,
if in the expression of the lineolinear invariant ac' — 266' + ca', we put the two quadrics
equal to each other, we have the quadrinvariant
ac— 6*
of the single quadric.
518 A FOURTH MEMOIR UPON QUANTICS. [155
70. The lineolineajr invariant ah* — o!h of two linear functions may be considered as
giving the lineoIinear covariant d^U .dyV—dyU . dgV of the two quantics U and F,
and in like manner the lineolinear invariant ac' — 2bb' + ca* may be considered as giving
the lineolinear covariant dx^U .dy^V—^dJdyU .dJdyV+dy^U .dx*V o{ the quantics U, V.
And generally, any invariant whatever of a quantic or quantics of a given order or orders
leads to a covariant of a quantic or quantics of any higher order or orders: viz. the
coefficients of the original quantic or quantics are to be replaced by the derived functions
of the quantic or quantics of a higher order or ordera
71. The same thing may be seen by means of the theory of Emanants. In tact,
consider any emanants whatever of a quantic or quantics; then, attending only to the
facients of emanation, the emanants will constitute a system of quantics the coefficients
of which are derived functions of the given quantic or quantics; the invariants of the
system of emanants will be functions of the derived functions of the given quantic or
quantics, and they will be covariants of such quantic or quantics; and we thus pass
from the invariants of a quantic or quantics to the covariants of a quantic or quantics
of a higher order or orders.
72. It may be observed also, that in the case where a tantipartite invariant, when
the several quantics are put equal to each other, does not become equal to zero, we may
pass back from the invariant of the single quantic to the tantipartite invariant of the
system ; thus the lineolinear invariant ac* — 266' + ca' of two quadrics leads to the quadiin
variant ac — 6* of a single quantic ; and conversely, from the quadnnvariant oc — 6* of a
single quadric, we obtain by an obvious process of derivation the expression cuf — 266' + ca'
of the lineolinear invariant of two quadrics This is in fact included in the more general
theory explained. No. 64.
73. Reverting now to binary quantics, two quantics of the same order, even or odd,
have a lineolinear invariant. Thus the two quadrics
(a, 6, c^x, yy, (a\ h\ c'^a?, yf
have (it has been seen) the lineolinear invariant
acf  266' + ca* ;
and in like manner the two cubics
(a, 6, c, djir, y)», (a', 6', c', d'$a:, yf
have the UneoUnear invariant
ad!  36c' + 3c6'  da\
which examples are sufficient to show the law:
74. The lineolinear invariant of two quantics of the same odd order is a combinant,
but this is not the case with the lineolinecur invariant of two quantics of the same even
order. Thus the lastmentioned invariant is reduced to zero by each of the operations
155] A FOURTH MEMOIR UPON QUANTICS. 519
and
but the invariant
is by the operations
and
reduced respectively to
and
afda + b'db + c'dc + cfda ;
ac' 266' 4 ca'
a'da + Vdb + c'dc
2(ac  6^
2(aV6'0.
75. For two quantics of the same odd order^ when the quantics are put equal to
each other, the lineolinear invariant vanishes; but for two quantics of the same even
order, when these are put equal to each other, we obtain the quadrinvariant of the single
quantia Thus the quadrinvariant of the quadric (a, 6, c^x, yY is
oc — 6*;
and in like manner the quadrinvariant of the quartic (a, 6, c, (2, e\x, yY is
oe  46d + 3c».
76. When the two quantics are the first derived functions of the same quantic
of any odd order, the lineolinear invariant does not vanish, but it is not an invariant
of the single quantic. Thus the Uneolinear invariant of
(a, 6, c$a?, yy
and
(6, c, d^x, yy
is
(ad — 26c f c6 = ) od — 6c,
which is not an invariant of the cubic
(a, 6, c, d$a?, y)".
But for two quantics which are the first derived functions of the same quantic of
any even order, the lineolinear invariant is the quadrinvariant of the single quantic.
Thus the lineolinear invariant of
(a, 6, c, d^x, yy
and
(6, c, d, e^x, yy
is
(a«  36d + 3c"  d6 =) a«  4f6(i + 3c*,
which is the quadrinvariant of the quartic
(a, 6, c, d, e$x, yy.
520 A FOURTH MEMOIR UPON QUANTICS. [155
77. I do not stop to consider the theory of the lineolinear covariants of two
quanties, but I derive the quadricovariants of a single quantic directly fix>m the
quadrinvariant. Imagine a quantic of any order even or odd. Its successive even
emanants will be in regard to the facients of emanation quantics of an even order,
and they will each of them have a quadrinvariant, which will be a quadricovariant of
the given quantic. The emanants in question, beginning with the second emanant, are
(in regard to the facients of the given quantic assumed to be of the order m) of the
orders m — 2, m — 4,... down to 1 or 0, according as m is odd or even, or writing
successively 2p+l and 2p in the place of m, and taking the emanants in a reverse order,
the emanants for a quantic of any odd order 2pf 1 are of the orders 1, 3, 5... 2p — 1,
and for a quantic of any even order 2p, they are of the orders 0, 2, 4 ... 2p— 2. The
quadricovariants of a quantic of an odd order 2p + 1, are consequently of the orders
2, 6, 10...4p — 2, and the quadricovariants of a quantic of an even order 2p, are of
the orders 0, 4, 8 ... 4p — 4. We might in each case carry the series one step further,
and consider a quadricovariant of the order 4p f 2, or (as the case may be) 4p, which
arises from the 0th emanant of the given quantic; such quadricovariant is, however,
only the square of the given quantic.
78. In the case of a quantic of an evenly even order (but in no other case) we
have a quadricovariant of the same order with the quantic itsel£ We may in this
case form the lineolinear invariant of the quantic and the quadricovariant of the same
order: such lineolinear invariant is an invariant of the given quantic, and it is of
the degree 8 in the coefficients, that is, it is a cubinvariant. This agrees with the
beforementioned theorem for the number of cubinvariants.
79. In the case of the quartic (a, 6, c, d, e$a?, y)*, the cubinvariant is, by the
preceding mode of generation, obtained in the form
c(ac 6») 4dJ(ad 6c)f 6cJ (oe 46d + 3c«) 46 (becd) + a(ce (?),
which is in ^t equal to
3 (ace acP b^e f 2bcd  c») ;
and omitting the numerical factor 3, we have the cubinvariant of the quartic
•
80. In the case of a quantic of any order even or odd, the quadrinvariants of the
quadricovariants are quartinvariants of the quantic. But these quartinvariants are not
all of them independent, and there is no obvious method grounded on the preceding
mode of generation for obtaining the number of the independent (asyzygetic) quartin
variants, and thence the number of the irreducible quartinvariants of a quantic of a
given order.
81. I take the opportunity of giving some additional developments in relation to
the discriminant of a quantic
(a, 6, ...6\ a'][a?, y)'».
To render the signification perfectly definite, it should be remarked that the discrimiuant
contains the term a"*^^a"*^^ and that the coefficient of this term may be taken to be
155] A FOURTH MEMOIB UPON QUANTICS. 521
+ 1. It was noticed in the Introductory Memoir, that, by Joachimstharjs theorem, the
discriminant, on putting a = 0, becomes divisible by 6', and that throwing out this
factor it is to a numerical factor pris the discriminant of the quantic of the order
(w— 1) obtained by putting a = and throwing out the factor x] and it was also
remarked, that this theorem, combined with the general property of invariants, afforded
a convenient method for the calculation of the discriminant of a quantic when that
of the order immediately preceding is known. Thus let it be proposed to find the
discriminant of the cubic
(a, b, c, d^x, yy.
Imagine the discriminant expanded in powers of the leading coefficient a in the form
then this function. gu^l invariant must be reduced to zero by the operation 369a + 2cdb +dde;
or putting for shortness V = 2<^t + dde, the operation is V + Sbda, and we have
a^VA+aVB +V(7l
^ = 0,
f a 664 f SbBj
and consequently
But C is equal to 6* into the discriminant of (36, 3c, d^x, y)*, that is, its value is
6'(126(i— 9c'), or throwing out the factor 3, we may write
C = 46»d36»c«;
this gives
£ =  ^^( 66»cd + 246»cd 126c*),
or reducing
^ = 66cd + 4c»;
and thence
il =  ^ ( 66d* + Uc'd  12c»d),
or reducing
A=d\
which verifies the equation VA = 0, and the discriminant is, as we know,
a«d«  6a6cd f 4ac» + 46»d  36V.
82. If we coiisider the quantic (a, 6, ...a$jr, 1)** as expressed in terms of the
roots in the form a(x 'ay)(x  /3i/)..., then the discriminant (= a'""^ a *""* + &c. as
above) is to a factor prh equal to the product of the squares of the differences of
the roots, and the factor may be determined as follows: viz. denoting by f(a, /S, ...)
the product of the squares of the differences of the roots, we may write
a*"» ? (a, 13, . . . ) = AT (d^^ a'"*"* + &c.),
c. II. G6
522
A FOUBTH MEMOIB UPON QUANTIC8.
[155
where ^ is a number ; and then considering the equation a;*" — 1 = 0, we have to
determine N the equation
But in general
and if
then
or
here
and therefore
but
or
and
whence
or
and consequently
{:(«. /9, .;.) = ()•»■ .V.
<f>x = (x — a) (x — 0) ...,
(a/3)(a7)... = <^'a, &c.,
?(a, /3, ...) = ()*"*<"''» f«f/3...;
(ra/87... = l,
a/37...=()«U,
J^ = ()4nr(mi) ^,n^
a*"* ?(«, /9, . ; .) = ()*'"'"»>' m'" (a"*» a "»' + &c.),
or what is the same thing, the value of the discriminant D (=a'*"*a'*~* + &c.) is
()Jm(mi)^mamjf(a, /8, ...)•
It would have been allowable to define the discriminant so as that the leading term
should be
m
in which case the discriminant would have constantly the same sign as the product
of the squared differences; but I have upon the whole thought it better to make
the leading term of the discriminant always positive.
83. A quautic of an even order 2p has an invariant of peculiar simplicity, viz.
the determinant the terms of which are the coefficients of the pih differential
coefficients, or derived functions of the quantic with respect to the facients ; such
invariant may also be considered as a tantipartite invariant of the pih emanaubi.
Thus the sextic
(a, 6, c, (/, e, /, flr$d?, yY
155] A FOUBTH MEMOIR UPON QU ANTICS. 523
has for one of its invariante, the determinant
\ a, b, c, d ,
\ b, Cy d, e \
c , cf , e , f
d^ ey f , g
The invariant in question is termed by Professor Sylvester the Catalecticaut.
84. Professor Sylvester also remarked, that we may from the eatalecticant form
a function containing an indeterminate quantity X, such that the coefficients of the
diflFerent powers of X are invariants of the quantic; thus for the sextic, the function
in question is
a y b ,c ,d— \
b , c , cf + JX, c
, c , d  JX, e , /
d+\ e , f y g
where the la»v of formation is manifest; the terms in the sinister diagonal are
modified by annexing to their numerical submultiples of X with the signs + and —
alternately, and in which the multipliers are the reciprocals of the binomial coefficients.
The function so obtained is termed the Lambdaie,
85. If we consider a quantic of an odd opder, and form the eatalecticant of the
])enultimate emanant, we have the covariant termed the Canonisant. Thus in the case
of the quintic
(a, 6, c, dy ey f \x, y)\
the canonisant is
j ax{bg, bx + cy, cx + dy \
\ bx { eg, cx^dyy dx^eg .\
' ex H dy, dx+ «y, ex vfg
which is equivalent to
a , 6 , c , d
by C y d y €
C , d y e y / ,
and a like transformation exists with respect to the canonisant of a quantic of any
odd order whatever. The canonisant and the lambdaie (which includes of course the
•
eatalecticant) form the basis of Professor Sylvesters theory of the Canonical Forms
of quantics of an odd and an even order respectively.
66—2
524 A FOURTH MEMOIR UPON QUANTIC8. [l55
86. There is another family of covariants which remains to be noticed. Consider
any two quantics of the same order,
(a, 6,...$a?, y)"»,
(a, 6', ...$a?, y)~,
and join to these a quantic of the next inferior order,
where the coefficients {u, v, ...) are considered as indeterminate, and which may be
spoken of as the adjoint quantic.
Take the odd iineoiinear covariants (viz. those which arise from the odd emanants)
of the two quantics; the term arising from the (2i + l)th emanants is of the form
where (4, jB, ...) are lineolineai* functions of the coefficients of the two quantics.
Take also the quadricovariants of the adjoint quantic; the term arising from the
(2i — m)th emanant is of the form
where {U, F, ...) are quadric functions of the indeterminate coefficients (u, v, ...). We
may then form the quadrin variant of the two quantics of the order 2(m — 1 — 2i):
this will be an invariant of the two quantics and the adjoint quantic, lineolinear in
the coefficients of the two quantics and of the degree 2 in regard to the coefficients
(m, v, ...) of the adjoint quantic; or treating the lastmentioned coefficients as iacient«,
the result is a lineolinear mary quadric of the form
(a, aB,...$u, r,...)«,
viz. in this expression the coefficients J3[, J3, ... are lineo linear functions of the co
efficients of the two quantics. And giving to % the different admissible values, viz.
from t = to i = ^m — 1 or ^^(m — 1) — 1, according as m is even or odd, the number
of the functions obtained by the preceding process is Jm or ^ (m — 1), according as
m is even or odd. The functions in question, the theory of which is altogether due
to Professor Sylvester, are termed by him Cohezoutiants ; we may therefore say that
a cobezoutiant is an invariant of two quantics of the same order m, and of an adjoint
quantic of the next preceding order w — 1, viz. treating the coefficients of the adjoint
quantic as the facients of the cobezoutiant, the cobezoutiant is an 7Hary quadric, the
coefficients of which are lineolinear functions of the coefficients of the two quantics,
and the number of the cohezoutiants is ^i or ^(m — 1), according as m is even or
odd.
87. If the two quantics are the differential coefficients, or first derived functions
(with respect to the facients) of a single quantic
(a, 6, ...$a?, y)*~.
155]
A FOUBTH MEMOIR UPON QUANTICS.
525
then we have what are termed the Cohezoutoids of the single quantic, viz. the cobe
zoutoid is au invariant of the single quantic of the order m, and of an adjoint quantic
of the order (m — 2) ; and treating the coefficients of the adjoint quantic as facients,
the cobezoutoid is an (wi — l)ary quadric, the coefficients of which are quadric functions
of the coefficients of the given quantic. The number of the cobezoutoids is ^(m — 1)
or ^ (m — 2), according as m is odd or even.
88. Consider any two quantics of the same order,
(a, . . .$a?, y)*~, (a', ...v$a?, y)*",
and introducing the new facients (X, F), form the quotient of determinants,
(a, ...$0?, y r, {a,.,.^x , y)
^ , y
X,Y
which is obviously an integral function of the order {m — 1) in each set of facients
separately, and lineolinear in the coefficients of the two quantics; for instance, if the
two quantics are
(a, 6, c, d^x, yy,
{a\ h\ c\ d'^x, yY,
the quotient in question may be written
( 3 (at'  a'b\ 3 [ac'  a'c) , ad'  a!d \x, y)« {X, Y)\
3 {ac'  a'c), ad'  a'd + 9 Q>c'  6'c), 3 {bd'  b'd)
ad'a'd, Sibd'b'd) , 3 {cd' ^ c'd)
The function so obtained may be termed the Bezoutic Emanant of the two quantics.
89. The notion of such function was in fact suggested to me by Bezout s abbre
viated process of elimination, viz. the two quantics of the order m being put equal to
zero, the process leads to (w — 1) equations each of the order (m — 1): these equations
are nothing else 'than the equations obtained by equstting to zero the coefficients of
the different terras of the series {X, F)*^* in the Bezoutic emanant, and the result
of the elimination is consequently obtained by equating to zero the determinant
formed with the matrix which enters into the .expression of the Bezoutic emanant.
In other words, this determinant is the Resultant of the two quaiitic& Thus lAie resultant
of the lastmentioned two cubics is the determinant
3(a6'ai), 3(ac'a'c) , ad'a'd
3(ac'a'c), ad!  a'd + 9 (be'  b'c), 3(6d'6'rf)
ad'a'd, 3bd'b'd , 3(cd'c'rf)
526
A FOURTH MEMOIR UPON QUANTICS.
[155
90. If the two qualities are the dififerential coeflScients or first derived functions
(with respect to the facients) of a single qiiantic of the order m, then we hare in
like manner the Bezoutoidal Emanant of the single quantic; this is a function of the
order (m — 2) in each set of facients, and the coefficients whereof are quadric functions
of the coeflBcients of the single quantic. Thus the Bezoutoidal emanant of the quartic
(a, 6, c, d, e\x, yY
IS
( 3(ac6»), 3 (ad be) , aebd 5^, y)^(X, Yf
3(ad6c), ae + SbdQd', S(b€cd)
ae — bd, 3(b€ — cd) , 3 (ce  d* ) ,
and of course the determinant formed with the matrix which enters into the expression
of the Bezoutoidal Emanant, is the discriminant of the single quantic.
91. Professor Sylvester forms with the matrix of the Bezoutic emanant and a set
of m facients («i, v, ...) an mary quadric function, which he terms the Bezoutiant
Thus the Bezoutiant of the before mentioned two cubics is
i 3 {aV  a'6), 3 {ac'  a'c) , ad'  a'd ^w, v, wf ;
i 3(ac'ac), ad'  a'd 4 9 (6c'  6'c), 3(6d'6'd)
I ad'  a'd , 36d'  6'd ,3 (cd'  c'd) ,
and in like manner with the Bezoutoidal emanant of the single quantic of the order wi
and a set of (m — 1) new facients (w, v, ...), an (m— l)ary quadric function, which he
terms the Bezoutoid. Thus the Bezoutoid of the beforementioned quartic is
( 3 (ac — 6'), 3 (ad — be) , ae — bd $w, v, w)\
3 (ad  6c), ae 4 86d  9c», 3 {be cd)
ae —bd, 3{be — cd), 3 (ce — d*)
To him also is due the important theorem, that the Bezoutiant is an invariant of
the two quantics of the order m and of the adjoint quantic (u, v, ...$y, — a:)**~S being in
fact a linear function with mere numerical coefficients of the invariants called Cobe
zoutiants, and in like manner that the Bezoutoid is an invariant of the single quantic
of the order m and of the adjoint quantic {u, v, ...$y, — a?)"*"*, being a linear function
with mere numerical coefficients of the invariants called Cobezoutoids.
The modes of generation of a covariant are infinite in number, and it is to be
anticipated that, as new theories arise, there will be frequent occasion to consider new
processes of derivation, and to single out and to define and give names to new co
variants. But I have now, I think, established the greater part by far of the definitions
which are for the present necessary.
1561
527
156.
A FIFTH MEMOIR UPON QUANTICS.
[From the Philosophical Transactions of the Royal Society of London, vol. cxLViii. for
the year 1858, pp. 429—460. Received February 11,— Read March 18, 1858.]
The present memoir was originally intended to contain a development of the
theories of the covariants of certain binary quantics, viz. the quadric, the cubic, and
the. quartic; but as regards the theories of the cubic and the quartic, it was found
necessary to consider the case of two or more quadrics, and I have therefore com
prised such systems of two or more quadrics, and the resulting theories of the
harmonic relation and of involution, in the subject of the memoir; and although the
theory of homography or of the anharmonic relation belongs rather to the subject of
bipartite binary cjuadrics, yet from its connexion with the theories just referred to, it
is also considered in the memoir. The paragraphs are numbered continuously with
those of my former memoirs on the subject : Nos. 92 to 95 relate to a single quadric ;
Nos. 96 to 114 to two or more quadrics, and the theories above referred to; Nos.
115 to 127 to the cubic, and Nos. 128 to 145 to the quartic. The several quantics
are considered as expressed not only in terms of the coefficients, but also in terms of
the roots, — and I consider the question of the determination of their linear factors, —
a question, in effect, identical with that of the solution of a quadric, cubic, or
biquadratic equation. The expression for the linear factor of a quadric is deduced
from a wellknown formula; those for the linear factors of a cubic and a quartic
were *fir8t given in my " Note ' sur les Covariants d'une fonction quadratique, cubique
ou biquadratique k deux indc^termin^es," Crelle, vol. L. (.1855), pp. 285 — 287, [135]. It is
remarkable that they are in one point of yiew more simple than the expression for
the linear factor of a quadric.
92. In the case of a quadric the expressions considered are
(a, 6, cjx, y)\ (1)
ac6« , . (2)
528 A FIFTH MEMOIR UPON QU ANTICS. [l56
where (1) is the quadric, and (2) is the discriminant, which is also the quadrin variant,
catalecticant, and Hessian.
And where it is convenient to do so, I write
(2) = D.
93. We have
(Be,  36, da'lx, yy D = tr,
which expresses that the evectant of the discriminant is equal to the quadric ;
(a, 6, cja^, a^)« 1/^=40,
which expresses that the provectant of the quadric is equal to the discriminant ;
(a, 6, c^bx + cy, —ax — hyf = D tT,
which expresses that a transmutant of the quadric is equal to the product of the
quadric and the discriminant.
94. When the quadric is expressed in terms of the roots, we have
a^ U = (xay)(X'fiy\
and in the case of a pair of equal roots,
a''U^(xOLyy,
D =0.
95. The problem of the solution of a quadratic equation is that of finding a
linear factor of the quadric. To obtain such linear factor in a symmetrical form, it
is necessary to introduce arbitrary quantities which do not really enter into the solution,
and the form obtained is thus in some sort more complicated than in the like
problem for a cubic or a quartic. The solution depends on the linear transformation
of the quadric, viz. if we write
(a, 6, c$\x + fiy, vx + pyy = (a', h\ c'Ja?, y)*,
80 that
a'— (a, 6, c$X, v)\
V = (a, 6, c$X, i/J/A, p),
c' = (a, 6, c\fi, p)»,
then
a'c  6'« = {ac  ¥) (\p  fipy,
an equation which in a different notation is
(a, 6, cja:, y)«.(a, 6, c$Z, Yy{(a, h c^x, yJZ, F))^ = D (F*Xy)«.
156]
A FIFTH MEMOIR UPON QUANTICS.
529
in which form it is a theorem relating to the quadne and its first and second
emanant& The equation shows that
(a, h, clx, ylX, F) + V rn ( Kr  Zy),
where (X, Y) are treated as supernumerary arbitrary constants, is a linear factor of
(a, 6, c\xy yy, and this is the required solution.
96. In the case of two quadrics, the expressions considered are
(a, 6, c'^x, y)\ (1)
(a', h\ c'\x, y)\ (2)
ach" , (3)
ac'266'4ca', (4)
6" , (5)
ac
> »
(6)
(o6'  a'b,
(Xa + fia',
ac —ac
be'  Vc \x. yf,
\c+nc' ^x, yy,
{ac b* , ac' 266' + ca', a'c'  6'» \\, ft)',
(7)
(8)
(9)
(1) and (2) are the quadrics, (3) and (5) are the discriminants, and (4) is the lineo
linear invariant, or connective of the discriminants ; (6) is the resultant of the two
quadrics, (7) is the Jacobian, (8) is an intermediate, and (9) is the discriminant of the
intermediate. And where it is convenient to do so, I write
(1) = u,
(2) = v.
(3) = D,
(*) = Q.
(5) = □'.
(6) = R.
(7) = H.
(8) = W.
(9) = e.
C. II.
530
A FIFTH MEMOm UPON QUANTIC8.
97. The Jacobian (7) may also be written in the form
a, b , c
The Resultant (6) may be written in the form
[156
a,
26.
a, 26,
c.
a\
26'.
a', 26',
c'.
and also, taken negatively, in the form
4 {ab'  a'6) (6c'  Vc)  {ac'  a'c)»,
which is the discriminant of the Jacobian ; and in the form
4 (oc  ¥) (a V  6'»)  {ac'  2hV + caj,
which is the discriminant of the Intermediate.
98. We have the following relations:
(a, 6, c\Vx + c'y,  a'x  6'y)» =  (aV  6'*) (a , 6 , c\x, yf
+ (ac'266'4ca') (a', 5', c'$a?, y)«,
(a', 6', c' \hx \cy,ax byY = + (oc'  266' + ca') (a , 6 , c $a?, y)»
(ac6«) (a', 6', c'l^x, y)\
and moreover
(ac6«, ac' 266' 4 ca', a'c'6'«$£r,  tO*
=  {{ab'  a'6, oc'  a'c, 6c'  6'c][a', y)*}*,
an equation, the interpretation of which will be considered in the sequel.
"99. The most important relations which may exist between the two quadrics are
First, when the connective vanishes, or
ac'  266' + ca' = 0,
in which case the two quadrics are said to be karmonically related: the nature of
this relation will be further considered.
156] A FIFTH MEMOIR UPON QU ANTICS. 531
Secondly, when i2 = 0, the two quadrics have in this case a common root, which
is given by any of the equations,
= 3ii'B : dff'R : d^R
= 6c' — Vc : ca' — c'a : aV — a'6.
The last set of values express that the Jacobian is a perfect square, and that
the two roots are each equal to the common root of the two quadrics.
The preceding values of the ratios a^ : 2xy : y' are consistent with each other in
virtue of the assumed relation i2 = 0, hence in general the functions
4SaR . dcR  (djty, daR . db'R  dbR . daR, &c.
all of them contain the Resultant i2 as a factor.
It is easy to see that the Jacobian is harmonically related to each of the quadrics;
in fact we have identically
a(6c'6'c)46 (ca'c'a)4c (a6'a'6) = 0,
a' (6c'  6'c) + V {ca'  c'a) 4 c' (a6'  a'6) = 0,
which contain the theorem in question.
100. When the quadrics are expressed in terms of the roots, we have
a^U =(a?ay)(a?^y),
a''W =(^a'y)(j:^'y),
4a^n =(a)8)«,
2 (aa')' = 2a^ + 20^)8' {a ^13) (a' + /S'),
4a'^n' =(a'^')»,
(aa^^R =(aa')(a^')()8a')(/S/8'),
(oaT'H =
y«, 2yx , x^
1, a+^, afi
1, a' + iS', o^ff
101. The comparison of the lastmentioned value of K with the expression in
terms of the roots obtained from the equation
ij=4nn'Q»,
gives the identical equation
which may be easily verified.
67—2
532
A FIFTH MEMOIR UPON QUANTIC8.
[156
102. We have identically
2a/8 + 2a'/8'  (a + yS) (o' + /S')
= 2 (a «')(« /SO (a /9)(2a
= 2(/3a')(y8 ^)(fi a)(2ff
= 2(0' a)(a'  /3 )  (a'  /T) (2a'
= 2 09'  a ) (/y  yS )  (/3'  0') (2/3'
a:
a'
a
a
/9);
and the equation Q^^ac' — 266' + ca' = . njay consequently be written in the several
forms
a/S aa' afi
2 1 1
/ >
/3a aa' ' fi'fi'
1 1
+ r
a'/3' a'a ' o'/8'
1 1
+
so that the roots (a, yS), (a', /3') are harmonically related to each other, and hence the
notion of the harmonic relation of the two quadrics.
103. In the case where the two quadrics have a common root a = a',
a'
U
= (^
ay) {x 
/8y),
a''
U'
= (x
ay) {x 
/9'y).
4a*n
= (a
/8)'.
2 (aa')'Q
= («
8)(a
■n
4a'
,□'
= (a
yS')'.
R
= 0,
(m')' H = (^  0) (X  a^y.
104. In the case of three quadrics, of the expressions which are or might be
considered, it will be sufficient to mention
(a , b , c Ijix, yY,
(a ,h',c' $ar, yf.
(a", b", c"\x, yy.
a , b , c ' >
a , , c
a", b", c"
(1)
(2)
(3)
(4)
156]
A FIFTH MEMOIR UPON QU ANTICS.
533
where (1), (2), (3) are the quadrics themselves, and (4) is an invariant, linear in the
coefficients of each quadric. And where it is convenient to do so, I write
(1) = u,
(2) = W,
(3) = U".
(4) = n.
105. The equation H = is, it is clear, the condition to be satisfied by the
coefficients of the three quadrics, in order that there may be a syzygetic relation
\U + fjkU' + vU" = 0, or what is the same thing, in order that each quadric may be
an intermediate of the other two quadrics; or again, in order that the three quadrics
may be tn Involution. Expressed in terms of the roots, the relation is
1, a +13, afi
1, af+^, a[ff
1, a" + ^", a";8
n Qii
= 0;
and when this equation is satisfied, the three pairs, or as it is usually expressed, the
six quantities % fi\ a!, ff \ ol\ ff\ are said to be in involution, or to form an
involution. And the two perfectly arbitrary pairs a, )8; oi^ ff considered as belonging
to such a system, may be .spoken of as an involution. If the two terms of a pair
are equal, e.g. if a" = ff' = ^, then the relation is
1, 2^ , ^
1, a+)8, dfi
1, o' + ^S', alff
= 0;
and such a system is sometimes spoken of as an involution of five terms. Con
sidering the pairs (a, fi\ (a', )8') as given, there are of course two values of 6 which
satisfy the preceding equation; and calling these 6^ and 6^^, then 6, and 0,^ are said
to be the sibiconjugates of the involution a, ^8; a', ff. It is easy to see that ^^, 6^^
are the roots of the equation H=0, where H is the Jacobian of the two quadrics
U and IT whose roots are (ot, /8), («', ff). In fact, the quadric whose roots are 0,y 0„ is
y\ 2yx , od"
1, a+)8, ap
1, a' + ^, a'^
which has been shown to be the Jacobian in question. But this may be made clearer
as follows: — If we imagine that X, fi are determined in such manner that the inter
mediate \V ¥ fiir may be a perfect square, then we shall have \U'{fiir — a''{x'0yY,
where denotes one or other of the sibiconjugates 0^, 0^^ of the involution. But the
condition in order that \U •{■ filT may be a square is
{ac  h\ ad  266' + ca, dd  y'\\ fiy ;
534
A FIFTH MEMOIR UPON QUANTICS.
[156
and observing that the equation \ : fi= W : ^U implies \U+ filT ^0=^a'' (x— OyY, it
is obvious that the function
must be to a fetctor prh equal to (x — O^yf (x — 0„yy. But we have identically
(ac  6«, ac'  266' + ca\ ale'  V'^W, ^U)'^^ {{ab'  a% ac'  a'c, be'  Vcl^x, y)»}«,
and we thus see that (x — 0^y), (x — 0^^y) are the £Eu;tors of the Jacobian.
106. It has been already remarked that the Jacobian is harmonically related to
each of the quadrics U, U'; hence we see that the sibiconjugates 5,, 0„ of the
involution a, y3, a^, ff are a pair harmonically related to the pair a, /3, and also
harmonically related to the pair a\ ^, and this properly might be taken as the
definition for the sibiconjugates 5,, 0,^ of an involution of four terms. And moreover,
a, P\ a\ fi' being given, and 0^, 0^^ being determined as the sibiconjugates of the
involution, if a", /S^' be a pair harmonically related to 0,, 5,,, then the three pairs
a, /9; a', )8'; a", /8" will form an involution; or what is the same thing, any three
pairs a, /8; oi, ^\ a", ^", each of them harmonically related to a pair 0,, 6^^, will be
an involution, and 0^ , 0^^ will be the sibiconjugates of the involution.
107, In particular, if a, yS be harmonically related to 0^, 5,^, then it is easy to
see that ^^, 0^ may be considered as harmonically related to d^, d^^, and in like manner
^//» ^// wi^' ^^ harmonically related to 5,, 0^/, that is, the pairs 5,, 0/^ 0^^, 0^^ and
a, fi will form an involution. This comes to saying that the equation
1, ^e, . e; =0
1, a + /3, «^
is equivalent to the harmonic relation of the pairs a, /S; B^, 0^/, and in &ct the deter
minant is
(d,  0„) {2afi + 20,0,, (a + 13) (0, + 0,,)),
which proves the theorem in question.
108. Before proceeding further, it is proper to consider the equation
= 0,
1, a,
a',
aal
1. fi.
^,
^ff
1. 7.
i.
r/
1, 8.
8'.
^
which expresses that the sets (a, yS, 7, S) and (a', ff, y\ S^ are homographic; for
although the homographic equation may be considered as belonging to the theory of
156] A FIFTH MEMOIR UPON QU ANTICS. 535
the bipartite quadrie {x — ay) (x — ay), yet the theory of involution cannot be completely
discussed except in connexion with that of homography. If we write
^ =(/S7)(«8), 5=(7a)(^8), C ={a  p)iy i),
A' = (^Y)(c^n B' = Wc^)(B'n (7 = (a'  jSO (7'  8'),
then we have
and thence
BCFC^CA''C'A^AR^A'B;
and either of these expressions is in fact equal to the lastmentioned determinant, as
may be easily verified. Hence, when the determinant vanishes, we have
A : B : C^A' : R : C,
Any one of the three ratios A : B : C, for instance the ratio B : C,=
(7«)(/8"g)
(a)8)(78)' .
is said to be the anharmonic ratio of the set (a, ^, 7, S), and consequently the two
sets (a, ^, 7, S) and (a\ ff, 7', S') will be homographically related when the anharmonic
ratios (that is, the corresponding anharmonic ratios) of the two sets are equal.
If any one of the anharmonic ratios be equal to unity, then the four terms of
the set taken in a proper manner in pairs, will be harmonics; thus the etiuation
^ = 1 gives
which is reducible to
2aS 4. 2^7  (a + S) 08 + 7) = 0,
which expresses that the pairs a, B and 13, 7 are harmonics.
109. Now returning to the theory of involution (and for greater convenience
taking a, 0^ &c. instead of a, )8 &c. to represent the terms of the same pair), the
pairs a, a'; )8, iS'; 7, 7'; S, 8'; &c. will be in involution if each of the determinants
formed Mrith any three lines of the matrix
1, a +a! , aa' ,
1, fi + fi', pff,
^ 7+7» 77'»
1, S+S', SS',
&c.
536
A FIFTH MEMOIR UPON QUANTICS.
[156
vanishes: but this being so, the determinant
which is equal to
1,
a, a',
aa'
1.
A ^.
18/8'
1,
7. 7'.
77'
1,
8, 8',
SS'
«■
1, « + «'
, aa'
/9.
1, 13 + ^
> fi^
7.
1. 7+7'
. 77'
B,
1, B +B'
, SS'
will vanish, or the two sets (a, )8, 7, S) and (a', /S', 7', SO will be homographic ; that
is, if any number of pairs are in involution, then, considering four pairs and selecting
in any manner a term out of each pair, these four terms and the other terms of
the same four pairs form respectively two sets, and the two sets so obtained will be
homographic.
110. In particular, if we have only three pairs a, a'; fi, ff\ 7, 7', then the sets
a, )8, 7, a' and a', ff, 7', a will be homographic; in fact, the condition of homography is
which may be written
or what is the same thing,
1,
a,
a', aa!
=
}
1,
A
^. P^
1.
7.
7» Ti
1,
a'.
a, ojoi
a.
1,
a 4 a', aa'
J=
/9.
1,
/S + i8', P^
7.
1.
7 + 7'. 77'
a'.
1.
a +0', ew'
a
>
1, a + a* , oa'
/3
9
1. iS + yS', y9/8'
7
1
1. 7+7'. 77'
0'
a,
0,
. c
)
= 0,
so that the firstmentioned relation is equivalent to
(«'  «)
1, a + a' , ao*  = 0,
1, y8 + ^, /SyS
1. 7+7'. 77'
156]
A FIFTH MEMOIR UPON QUANTICS.
537
and the two sets give rise to an involution. The condition of homography as expressed
by the equality of the anharmonic ratios may be written
a7.a'^'"a'7'.a/8''
or multiplying out,
(a  ;8) (a  ;80 («'  7) (0'  yO  («'  ;8) (a'  )8') (a  7) (a'  7') = 0.
which is a form for the equation of involution of the three pairs. But this and the
other transformations of the equation of involution is best obtained by a different
method, as will be presently seen.
Ill, Imagine now any number of pairs a, a'; fi, /3^; 7, 7'; S, S'; &c. in involution,
and let x, y, z, w he the fourth harmonics of the same quantity X with respect to
the pairs a, a' ; fi, 13" ; 7, 7' and B, 8' respectively ; then the anharmonic ratios of the
set (Xy y, z, w) will be independent of X, or what is the same thing, if x\ y\ /, w
are the fourth harmonics of any other quantity X' with respect to the same four pairs,
the sets (a?, y, ^, w) and (x\ y', /, w') will be homographic, or we shall have
= 0.
1,
X,
x\
OCX
1,
y>
y'.
yy
1,
z.
«',
Z!f
1,
w,
w',
vm
It will be sufficient to show this in the ceese where X is anything whatever, but X'
has a determinate value, say X' = 00 ; and since if all the terms a, a', &c. are
diminished by the same quantity X the relations of involution and homography will
not be affected, we may without loss of generality assume X = 0, but in this case
X = — — , , a;' = i (a h a'),
and the equation to be proved is
1.
aa
1,
1,
1,
a + a"
77'
/ f
7+7
SB'
B + B^'
a f flf, aa* ; = 0,
/3 + /8', 13^
7+7'» . 77'
B + B\ BB'
which is obviously a consequence of the equations which express the involution of the
four pairs.
C. II. 68
538
A FIFTH MEMOIR UPON QUANTICS.
[156
A set homographic with x^ y, z w, which are the fourth harmonics of any quantity
whatever X with respect to the pairs in involution, a, a'; fi, 13'; 7, 7'; S, B\ is said to
be homographic with the four pairs, and we have thus the notion of a set of single
(quantities homographic with a set of pairs in involution. This very important theon
is due to M. Chasles.
112. Let r; 8] t he the anharmonic ratios of a set a, fi, 7, B, and let r/, s,; t,
be the anharmonic ratios (corresponding or not corresponding) of a set a^, ^^, 7^, h^ And
suppose that /; ^; t'; r/; <; e/; r"; *"; T; <; <'; C; ^'"; «'"; ^"; r/"; <"; C
are the analogous quantities for three other pairs of sets ; then an equation such as
TV, TV,
= 0.
or as it is more conveniently written,
»«, >
»•». .
»•/« .
»^,
«v ,
rv;
A" .
»^V' .
r/V .
«"V",
r'V.
r/'V",
r r,
=
is a relation independent of the particular ratios r : 8 which have been chosen for the
anharmonic ratios of the sets; this is easily shown by means of the equations
r + « + « = 0, r, + «, + «, = 0,
which connect the anharmonic ratios. The equation in fact expresses a certain relatiou
between four sets (a, ^, 7, S) and four other sets (a^, /8^, 7^, S^); a relation which may
be termed the relation of the homography of the anharmonic ratios of four and four
sets : the notion of this relation is also due to M. Chasles.
113. The general relation
1, a + /8 , a/3 =0
1, ot +^', f£ff
1, a" + i8", ot'P'
may be exhibited in a great variety of forms. In fact, if the determinant is denoted by
T, then multiplying by this determinant the two sides of the identical equation
w', — M, 1
v", V, 1
V^^ "W, 1
we obtain
T (u — v) (v — w) (w — w) =
= (w — v) (v — w) (w — u\
(Ma)(t//3), (t;a)(t;)8), (wa )(«; )8 ) !.
(^a')(u/9^), (t;a')(t;/}'), (w^a')(w^^)
(^0(^/9^), (t^a'Xt'rX {wa")(w^l3'')
156]
A FIFTH MEMOIR UPON QUANTICS.
539
If, for example, u = a, v = yS, then we have
T (a  ;8) =  (a  a') (a  /90 C9  O (/9  /8") + (/3  «') (/3  ^) («  «") («  /3") ;
and again, if « = a, v = a', v> = a", then we have
T =  (a  /3") (a'  ;8) (a"  i80 + (a  /S) («'  /9") («"  yS).
Putting T = Q, the two equations give respectively
(g  a') (ff  c^') _ («  yyp (/3  /y) .
(a«")(«'/8)~(«i8')(i9"/9)'
and
(«  /8") («'  /3) («"  /8') = (a  yS) («'  r) («"  /8),
which are both of them wellknown forma
114. A corresponding transformation applies to the equation
which expresses the homography o
representing by V the similar determinant
, a, of, aal =0,
, A ff. (iff
> 7» i> Tl
, &, O y CO
two pairs. In fact, calling the determinant 'V and
V9 =
ss' , — »' ,
s, 1
>
ttf, if.
t, 1
uu', — «',
tt. 1
m/ , —v'.
V. 1
ted to zero, would express the
we have
homography of
the sets («, t,
II, v) and
(««)(«'«'). {8fi)(8'n
(«7)(«'7).
(««)(*' 80
9
(t  a) (f  a'), {tfi)(1f n
(t  7) «'  7 ).
(<«)(«'«')
(«  a) («'  a!), (u  /S) (u'  ^,
(«  7) («'  y).
(m8)(m'S0
(va){t/af), (v
dywn
(»  7) (»
7).
(»  S) (t/  8')
which gives various forms of the equation of homography. In particular, if « = o^ s' = ff',
t = ff, (f = a', u = y, «' = S', » = S, v'=y, then
7^ =
(«
(/3
7)03'
7) (a'
70, («
70. (8
S)08'80
(7  «) (S'
(8  a) (y
«0. (7
/8)(8'
y8)(7'
/SO
68—2
540
A FIFTH MEMOIR UPON QUANTICS.
[156
and the righthand side breaks up into factors, which are equal to each other (whence
also V = '^), and the equation S?^ = takes the form
(«  7) (/8  S) («'  S') (/S  7')  («  8) (8  7) («'  7') (/S'  «') = 0.
which is, in fact, one of the equations which express the equality of the anharmonic
ratios of (a, )8, 7, S) and (a', ff, y\ S^.
115. In the case of a cul:)ic, tlie expressions considered are
(a, 6, c, d$a?, y)»,
(ac — b^ ad — be, bd — t^^a;, yf,
 a^d + %abc  26» '\
 abd + 2a(^  6«c
■\acd 2b^d 4 6c*
[ \ad'  Sbcd + 2c* ;
a«(?  6abcd + 4ac» 4 46»df  36V,
(1)
(2)
[^> y)**
(3)
(4)
where (1) is tlie cubic, (2) is the quadrico variant or Hessian, (3) is the cubico variant,
and (4) is the quartinvariant or discriminant.
And where it is convenient to do so, I write
(1)
(2)
(3)
(4)
U,
H.
so that we have
*»ni7»44JJ» = 0.
116. The Hessian may be written under the form
(cur + by)(cx\dy) — (6a? 4 cy)*,
(which, indeed, is the form imder which qua Hessian it is originally given), and under
the form
a, b ., c
6 , c , (Z
The cubicovariant may be written under the form
{2 (ac  6") a? + (ad  be) y] (6a^ + 2cxy 4 dy»)
 { {adbc)x + 2{bd  c")y} (cue* + 2&py 4 cy*),
156] A FIFTH MEMOIR UPON QU ANTICS. 541
that is, as the Jacobian of the cubic and Hessian ; and under the form
that is, as the evectant of the discriminant.
The discriminant, taken negatively, may be written under the form
+ 4 (oc  b^)(bd '(^){ad hcf,
that is, as the discriminant of the Hessian.
117. We have
(a, 6, c, d\ha^ + 2ca>y + df, aaf ihayy  c}/J = ir<^,
which expresses that a transmutant of the cubic is the product of the cubic and the
cubicovariant. The equation
{(3a, 9^, 3c, 9dl[», ^)'}«n=2Z7'
expresses that the second evectant of the discriminant is the square of the cubic.
The equation
# . 3cd , 36d + 6c« , 36ch2ad = 27 D'
3cd , 3c« +126d, 3ad66c , 3ac + 66«
36df6c» , 3arf66c , 36« + 12ac, 3aft
I
 36c  12ad,  3ac + 66' , 3a6 , a« '
expresses that the determinant formed with the second differential coefficients of the
discriminant gives the square of the discriminant.
The covariants of the intermediate aUh /3^ are as follows, viz.
118. For the Hessian, we have
H(aU^l3<P)= (1, 0, n3[a, /3yH
= (a>/8»n)ir;
for the cubicovariant,
^(a[7+^4>)= (0, D, 0, n» 5a, fifU
+ (1, 0, D, 05a,^)»c>
and for the discriminant,
Q(a£r + ^4>)= (1, 0, 2D, 0, D^^a, fi)*^
= (a«/S«n)«D,
where on the lefthand sides I have, for greater distinctness, written JV, &c. to denote
the functional operation of taking the Hessian, &c. of the operand aU + fi^,
542 A FIFTH MEMOIR UPON QUANTICa [156
In particular, if a = 0, ^ = 1,
119. Solution of a cubic equation.
The question is to find a linear factor of the cubic
(a, 6, c, d$a?, yY,
and this can be at once effected by means of the relation
between the covariants. The equation in fact shows that each of the expressions
is a perfect cube, and consequently that the cube root of each of these expressions
is a linear function of (x, y). The expression
is consequently a linear function of x, y, and it vanishes when [7 = 0, that is, the
expression is a linear factor of the cubic.
It may be noticed here that the cubic being a(a? — ay)(a? — ^y)(a? — 7y), then we
may write
^Ji^Tu^)  ^^(^U^U) = J a(a) tti>)(^ 7)(a?ay),
where 6> is an imaginary cube root of unity: this will appear firom the expressions
which will be presently given for the covariants in terms of the roots.
120. Canonical form of the cubic.
The expressions i(4>+ [/VD), i(4>— tTVO) are perfect cubes; and if we write
then we have
U^ x» + y»,
4) = Vn (x»  y»),
and thence also
ir=^nxy.
156]
A FIFTH MEMOIR UPON QUANTICS.
543
121. When the cubic is expressed in terms of the roots, we have
a^fT^ (x  ay){x  fiy)(x  7^) ;
and then putting for shortness
^=0ry)(a:ay), B ^ {r^ ■ a) {x  fiy\ C = (a  ^){x yy\
so that
Jl+£ + C = 0,
we have
a*^ =^^{B''C){C^A){A^B\
122. The covariants ff, 4> are most simply expressed as above, but it may be
proper to add the equations
a"
'a» + /8* + 7*  ^7  7a  aA
= ij 6a^7i87«7a»a)3»)8V7«aa«/3, jTo?, y^
=  4 {(a + ©i8 + ai«7) a? + (^7 + tt)7a + w'^aiS) y 1 {(« + ©'^/S h 0)7) a 4 (^7 4 ©V* + ®«i8) 3^]
(where © is an imaginary cube root of unity),
a^4>=^2(a^)(a7)«(a:^y)«(ar7y)
' 2(a« + ^' + 7')3(^ + 7a' + aiS" + )8»7 + 7*a + o?/3) + 12a^7,
2(a«/87 + /8Va + 7'a^) + 4(^' + 7»a« + a»^)(i87» + 7a« + a^ + ^7 + y^
 2 (a^V+)87'a*+7aW+4(a»^7+i8V+7*«i8)(^+7»aHa«/9'+^^ ' "^
^+2(^V + 7V + a»/3»)3(a)8V + )87V + 7a«/3» + a^V + ^7'«' + 7<3^i8") + 12^^ ;
= {(2a^~7)a;+(2^77a~a^)^} K2^7a)^+(27a ,7^^7)3^} {(27a^);r+(2a^/877a)yj
123. It may be observed that we have a^UU^ =  ^ A^B^O, which, with the
above values of H, 4> in terms of A, B, C and the equation A + BhG = 0, verifies
the equation 4>' — OU* + 4iH^ = 0, which connects the covariants. In fact, we have
identically,
{B^cyiCAyiABy^^
'4(A^B + CyABC^{A+BhC)^{BC^CA + ABy^lS{A+B^C)(BC{CA{AB)ABa
^(BC+CA+ABy 27 A'B'O,
by means of which the verification can be at once eflfected.
544 A FIFTH MEMOm UPON QUANTICS. [156
124. If, as before, cu is au imaginary cube root of unity, then we may write
27a»4> =(£C)(CA)(il~5),
27a» fT Vn = 3 (ft)  ft)«) ABC,
and these values give
27a»i(4)+ U'JD = [{a + 0)^ + arf) X ^ (By + a)^a + <oa^ ) y}\
27a» J (^  ^ ^3 = ((« + <»^ + ^'t) a? + (^87 + «7a + «'a/8) y}'»
and we thence obtain
^J(4>+£rVn)  ^i(4> [TVd) = _ Ja (ft>  fti')(i8  7)(ar  ay),
which agrees with a former result.
125. The preceding formulae show without difficulty, that each factor of the cubi
covariant is the harmonic of a factor of the cubic with respect to the other two factors
of the cubic; and moreover, that the factors of the cubic and the cubicovariant form
together an involution having for sibiconjugates the factors of the Hessian. In feet, the
harmonic of x — ay with respect to (^ /8y)(^ — 7y) is (2a — )8 7)0? + (2^7 — 7a — a/8)y,
which is a factor of the cubicovariant ; the product of the pair of harmonic factors is
(2a  ^  7)a:« + 2 087  a^)xy + ( 2a^7 + a^/S + a^) yM
and multiplying this by fi — y, and taking the sum of the analogous expressions, this
sum vanishes, or the three pairs form an involution. That the Hessian gives the sibi
conjugates of the involution is most readily shown as follows: — the lastmentioned
quadric may be written
((a + ^ + 7) + 3a)a;» + 2(a^4a74'i87a(a + ^ + 7>)a:y + (3a^7 + a(a)8 + a7 + /87))y*»
which is equal to
or, throwing out the factor 3a~S to
(6 + oa, 2c  26a, d f ca^x, y)\
which is harmonically related to the Hessian
(oc — 6^, ad — be, bd^c^^x, yY\
and in like manner the other two pairs of factors will be also harmonically related to
the Hessian.
156]
A FIFTH MEHOm UPON QUANTICS.
545
126. In the case of a pair of equal roots, we have
a'U=
(x  ayf {x  yy) ,
a^H =
 4 i<iyyix2yy,
a^ =
 lA («  7)* («  ay)*.
D =
0.
And in the case of all the roots equal, we have
H = 0, 4) = 0, n=o.
127. In the solution of a biquadratic equation we have to consider the cubic
equation tj'  Jlf (tj  1) = 0. The cubic here is (1, 0, —if, M\w, ly, or what is the
same thing,
(1. 0,  ii/, if $«r, 1)»;
the Hessian is
M(i. 1, iM'^w, 1)»;
the cubicovariant is
if(l, filf, p/, if+^ilf>$t!r, 1)»;
and the discriminant is
if«(l^if).
128.
In the case of a quartic, the expressions considered are
(a, 6, c, d, e^x, yY,
ac  iibd f Sc*,
(ac  6», 2 (ad 6c), ae\2hd Sc', 2 (6e  cd), ce d?\x, y)\
ace + 2bcd — ck? — J*6 — c*,
^ a«d+ 3a6c 2&», "j
 a»c  2 abd\ 9 oc*  6 6»c,
 5abe^ 15 acd  10 b^d,
f 10 ad«  10 6«g,
+ 5 ade+ 10 6d*  15 bee,
+ ae» + 2bde 9(fe \Gcd\
+ 6e«  3 c(fo + 2 d^*
^a^, y)*,
(1)
(2)
(3)
(4)
(5)
where (1) is the quartic, (2) is the quadrinvariant, (3) is the quadricovariant or Hessian,
(4) is the cubinvariant, and (5) is the cubicovariant.
C. II.
69
546 A FIFTH MEMOIR UPON QUANTICS. [l56
And where it is convenient to do so, I write
(1) = u.
(2) = / .
(3) = H.
(4) = /,
(5) = <t>.
The preceding covariants are connected by the equation
The discriminant is not an irreducible invariant, its value is
D = /'  27/» = a»e» + &c.,
for which see Table No. 12, [p. 272].
129. It is for some purposes convenient to arrange the expanded expression of the
discriminant in powers of the middle coeflBcient c. We thus have
D = aV  12 d'bde^  27 a«d*  6 ab^cfe  27 6V  64 6»d'
+ c ( 54 a*(?« + 54 a6V + 108 abd^ + 108 b'de)
+ c« ( 18 aV  180 abde + 36 6»d«)
+ c» ( 54 Ok?  54 b'e)
+ c* (81 ae).
130. Solution of a biquadratic equation.
We have to find a linear factor of the quartic
(a, b, c, d, e$x, yy.
The equation JU^ — lU^H \ AiH^^ — ^^, putting for shortness
may be written
(1, 0, M, M\IH, juy = \p^\
Hence, if w,, cr,, tsj are the roots of
(1, 0, if, Afl^tsr, 1)» = 0,
the expressions IH —miJU, IH — vr^JU, IH—vr^JU are each of them squares; write
(tjT,  tsr,) {IH  rff.JU) = X\
(tsr,  tsrO {IH  tsr, J(/) = 7«,
(tsy,  tsr,) (/^r  tsr, Ji/) = Z^ ,
156] A FIFTH MEMOIR UPON QU ANTICS. 547
so that, identically,
and consequently X iiY, X — lY are each of them squares. The expression
aX \ I3Y + yZ
will be a square if only
sua may be seen by writing it under the form
and in particular, writing Vtjj — btj, Vbt, — btj, Vbti — taj for a, )8, 7, the expression
is a square; and since the product of the different values is a multiple of U^ (this
is most readily perceived by observing that the expression vanishes for U = 0), the
expression is the square of a linear factor of the quartic.
131. To complete the solution: tjj, Wa, tj, are the roots of the cubic equation
(1, 0, iif, Jl/Jtsr, 1)» = 0;
and hence, putting for shortness,
P» = iif {(1, fif, JJlf, if+^if»$/^, Jf^)» + Vl,Vif(l, 0, JJlf, ilfj/if, JUf,
<?=iif{(l, ilf, lM,M+^iP^IH, JUy  s/T^^M {\, 0, Jif, 3/$//f, JfT)',
we have (o) being an imaginary cube root of unity)
i (®  «') («^«  «^3 ) (/^  «^i .^ fO = ^  ;
and if
Q,» = iif {lVl^i/},
then
J (w  ft)«) (iSTa  tsr,) = Po  Oo
Hence, multiplying and observing that (o) — cd')' = — 3, we find
 (^h^f ^"^  ^'^^ ^^^ ^xJU)=(p 0) (P.  0.).
and consequently
(w,  w,) 'JiHvfJU = (o)  6)») V  (P  Q) (P,  Q,).
We have, in like manner,
i(««')K«^.)(/^«r,Ji/)= ^ Q.
H«  «*) («.  ^i) iIHw,JU) = a>P »»Q,
i (w  «») («,  «,) (/fl^  nr,JU) = m*Pa> Q,
69—2
548 A FIFTH MEMOm UPON QU ANTICS. [156
and J (©  ai«) (isr,  tsr,) = Po  Oo,
4 (ft)  CD*) (tSr,  tSTi) = <» ^0  o^'Qo,
^ (ft)  O)') (iSTi  CTj) = ©'Po  O) Qo,
and therefore
(m,  t!r,) \/iir^;nT^ {<o  ft)») v (pq) Tj^v^Qo),
(tsr,  tsTj) V/fr "'ST^JU^ia)'' ft)») V  (ft)P  ft)^)(ft)Po  ft)«Q),
(«r, tsrO VZH^^^^;jT7 = (ft) ft)«) V  (ft)«P cdQ)(^«Pc  ft)Oo) ;
and hence disregarding the common factor &) — ft)^ the square of the linear factor of
the quartic is
V(P0)(PoOo) + V(ft)Pft)«(2)(ft>Poft)»(2i,) + V :r(^Pft)Q)(ft,='Poft)Qo),
which is the required solution.
It may be proper to add that
tJi= Po+ Oo,
— t32 = ft)Po+ft)*Qo,
 BTj = ft)'Po 4 ft) Oo.
132. The solution gives at once the canonical form of the quartic ; in fact, writing
X + tF= 2 ^(tJa  isr,) (tar,  bTi) VJ X^
Z  tF= 2 \/(t!r,'cr,)(isr,tsri) V7y^
where X, Y have their former significations, we find, by a simple reduction,
IE 'srJU^ (tsr,  isr,) •/ (x» + y«)^
/if  isr,Jfr=  (isr,  tsr,) /(x"  y'^,
igi^3Jtr=^^ ^^^^^^" ^V.4xy,
and thence putting
g^ tg, _ ^ (ft)  ft)') (ft)'Po + ft>Oo)
Wj — tJa (6)'Po — ««>0o)
we have
f;'=x* + y* + 6dxy,
which is the form required.
133. The Hessian may be written under the form
(9e, 9d, 9(j, 96, 9al[^, y)'«/,
that is, as the evectant of the cubinvariant.
156] A FIFTH MEMOm UPON QU ANTICS. 549
The cubicovariaut may be obtained by writing the qiiartic under the form
(aa? + 6y, bxicy, cx\dy, dx + ey'^x^ yy,
and then, treating the linear functions as coefficients, or considering this as a cubic,
the cubicovariaut of the cubic gives the cubicovariaut of the quartic.
If we represent the cubicovariaut by
4) = (a, b, c, d, e, f, g^x, y)\
then we have identically,
ag9ce + 8d»=0;
and moreover forming the quadrinavariant of the sextic, we find
ag 6bf + 15ce  10d» = JD,
where D is the discriminant of the quartic. From these two equations we find
bf4ce + 3d«=^n,
which is an expression given by Mr Salmon: it is the more remarkable as the left
hand side is the quadrin variant of (b, c, d, e, f $«?, y)*, which is not a covariant of the
quartic. It may be noticed also that we have
af3be + 2cd = 0,
bg  3cf 4 2de = 0.
134. The covariants of the intermediate
of the quartic and Hessian are as follows, viz.
The quadrinvariant is
/ (a£r + 6/8JT)= (/, 18/, 3/»5a, /8)«;
the cubinvariant is
J (aU + a^H)^ (J, I\ dIJ, /'h54J=3[a, /S)»;
the Hessian is
B{aU+6^H)= (1, 0, 3/5a, ^Y H
+ (0, /, 9/ 5a, ^yU:
and the cubicovariaut is
^{aU¥6^H)== (1, 0, 97, 54J5a, /3)»4>;
to which may be added the discriminant, which is
IIl(a£/' + 6/8if) = (l, 0, 187, 1087, 817^ 97277, 29167»l[a, ISyD.
550
A FIFTH MEMOm UPON QU ANTICS.
[156
135. The expression for the lambdaic is
6 , c + X, d
c — 2\, d , e
= J + X/4X=».
If the determinant is represented by A, that is if
then if Xi, Xj, \i are the roots of the equation A=0, and if the values of 3aA, &c
obtained by writing X, in the place of X are represented by 9a Aj, &c., then if x, y
satisfy the equation
(a, 6, c, d, e^x, y)* = 0,
we have identically (X, Y being arbitrary),
XyYx
+ V(9e, ad, Be, a^TaapTTyA;
a theorem due to Aronhold. I have quoted this theorem in its original form as au
application of the lambdaic, but I remark that
(a., Sd, 3c. 9&, da\X, 7yA = X(a,...$Z, F )*  (ac  6«, . . .^Z, Yy = \U'^H'
if J7', if' are what U, H become, substituting for (a?, y) the new facients {X, Y). More
over, we have
X = 
/ '
for substituting this value in the equation A = 0, we obtain the beforementioned equa
tion «j'— if (cr~l) = 0. We have, therefore,
(?e. 3d, 9<., 3», ^a\X, Yy K^^Xr H = j{IH' J^V),
and the equation becomes
( a, 6, c. d. ^.X F$g^y)^/37^ •JmJm,U''+ '^IH'Jwjr + 'JTU^J^.V
Xy — IX
156]
A FIFTH MEMOIR UPON QUANTICS.
551
Moreover, if {xay) be a factor of the quartic, then replacing in the formula y by the
value ax, (x, y) will disappear altogether; and then changing (X, Y) into (a?, y) where
X, y are now arbitrary, we have
(«, h, c. d, e^x, yf {a, 1 ) ^j^ ^ih^,JU + ^TR—^JU + ^llT^.W,
x — ay ' ^ '
which is a form connected with the results in Nos. 130 and 131.
136. We have
y*, 
ixy*.
6a?!/',
— 4a?i/, a?
= 6IHdJU;
a ,
36 .
3c , A
36 .
3c ,
d ,
6 ,
3c ,
M , e
3c ,
3d ,
e ,
a,
6.
it will appear from the formute relating to the roots of the quartic, that the ex
pression 6IH — 9JU vanishes identically when there are two pairs of equal roots, or
what is the same thing, when the quartic is a perfect square. The conditions in order
that the expression may vanish are obviously
6(acb^) : 3(ad6c) : ac + 26d3c» : 3(6ccd) : 6(ced«) : 9./
= r/ : i : C : d : e : I,
conditions which imply that the several determinants
6(ac6»), 3(ad6c), ae + 26d3c», 3 (6c erf), 6(ccd0
a , 6 , c , d , c
all of them vanish,
determinants are
we have identically
If for a moment we write 6H = (a\ h\ c\ d\ c'Ja, y)*, then the
j a', 6', c', d\ e i
a, 6, c, rf, e
ad' — a'd = 3 {he — 6 c),
eV  c'6 = 3 idc'  d'c).
a^'  a'c = 3 {hd'  h'd\
and the ten determinants thus reduce themselves to seven determinants only, these
in fact being, to mere numerical factors priSy the coefficients of the cubicovariant ;
this perfectly agrees with a subsequent result, viz. that the cubicovariant vanishas
identically when the quartic is a perfect square;
552
A FIFTH MEMOm UPON QUANTIC8.
[156
137. It may be remarked that the equation 6IH — 9JU = will be satisfied
identically if
a =
e =
c — <f>* c — <f)
where (f> !« arbitrary; the quartic is in this case the square of
(
Vc^' '^''^' v7^^*' ^^
If with the conditions in question we combine the equation 7 — (which in this cjise
ini[ilioh also .7 = 0), we obtain ^ = 0, and consequently
a __ i _ c _d
or the (juartic will be a complete fourth power.
It in easy to express in terms of the coeflBcients a, b\ c\ d\ e of 6H the dilicrent
dnterniiimnts
(i, 6, c, rf ,
6, c, d, 6 ;
wo bav(^ in fa(*t
aebd = ^(c+j VaV + 46'd'  3cA ,
8(6d c»)= i (c'  ^g V^vT4FJ'37^) ,
^
oc  6' = i a',
aci — 6c = J 6',
be —cd=^c\
[ ce (?=:Je',
wUi'.iu'ii all the abovementioned determinants will vanish, or the quartic will be a
fti*.vhtvX fourth power if only the Hessian vanishes identically.
138. Considering the quartic as expressed in terms of the roots, we have
a' U = (x ay) {x  fiy) (x  yy) (x  By)]
iiiA if we write for shortness
which are connected b}
C = (a/9)(78),
A+B + C = 0,
156]
A FIFTH MEMOIR UPON QUANTIOS.
553
then we have
a*I = ij(A* + B' + C*)'=r^{BC + CA+AB),
a*J=ji^(BC)iCA)iA B);
and for the discrimlDant we have
^^A'B'C,
and it is easy by means of a preceding formula to verify the equation □ = /' — 27J*.
139. The formulae show a very remarkable analogy between the covariants of a
cubic and the invariants of a quartic. In fact
For the cubic.
(A=(^y)(xtty),
fi = (ya)(a;/3y).
For the quartic.
'A = {By)iaB),
^ B=(ya)(BB),
.C = (a/3)(7«);
and then we have corresponding to each other:
For the cubic.
The Hessian,
The cubicovariant,
The cubic into the square root of the discriminant.
140. For the two covariants, we have
For the quartic.
The quadrinvariant.
The cubinvariant.
The discriminant.
and
if for shortness,
i3 = (S + i87a,
(B: = (S + rya/3,
141. We have
if=V
Bfi + ya, 8^(7+ a)7a (S + /8)$a;, y)»,
S7 + a/3, ^(a+/8)(^(S+7)$a:, y)».
(B'Cy(C'Ay(ABy'
or putting for shortness
we have
^(BC)(CA){A^B)'
M =
f(^» + £» + (?»)A«;
c. u.
70
554 A FIFTH MEMOIR UPON QUANTICS. [l56
and it is then easy to deduce
isri = A(£a),
in fact, these values give
W"! + «'a + t^s =0,
WltSTj f tSTjtJj f tJjtSTi ^ — iff,
'CJjtJjtS'i = ill,
and they are consequently the roots of the equation tx* ~ if(«r — 1) = 0.
142. The leading coefficient of IH—miJU is then equal to a* into the following
expression, viz.
3^(^« + £»+(?)a«(ac6«)7i^(A« + 5« + C*)(£C),
which is equal to
^(^« + £»+(?){48a>(ac6»)4(5C)},
and the term in { } is
8(fl^ + a7 + aS + ^7 + /8S + 7S)3(a + ^ + 7 + S)»4(7a)(/9S) + 4(a/8)(78),
which is equal to
3(S + a^7)».
But IH—'oriJU is a square, and it is easy to complete the expression, and we have
a^{IH  r!r,JU) = ~ ^frj (A'+ £. + O) j(g+^ « Y  a,  S^+ 7a, S/3 (^ + «)  ^ (S+/9)$a:, y)«}«,
We have, moveover,
«r, — «r8 = — 3Ail,
tT8""'ori = ""3A£,
and thence
x(S + a/87, Sa + /87, «« (iS + 7)  /97 (^ + «)$«, y)* ;
and taking the sum of the analogous expressions, we find
0* {(tsr,  tj,) •^IUvJU+ («r,  «,) 'JlH'atJU+ («r,  «•,) '^IHVfJU]
which agrees with a former result.
156] A FIFTH MEMOIR UPON QUANTICS. 555
143. The equation 7=0 gives
A : B : (7= 1 : c» : ©',
where cu is an imaginary cube root of unity; the factors of the quartic may be said
in this case to be Symmetric Harmonics,
The equation ^=0 gives one of the three equations,
A = B, B^C, C==A]
in this case a pair of factors of the quartic are harmonics with respect to the other
pair of factors. If we have simultaneously 7=0, «/=0, then
^=£ = = 0,
and in this case three of the fiskctors of the quartic are equal
144. If any two of the linear factors of the quartic are considered as forming,
with the other two linear factors, an involution, the sibiconjugates of the involution
make up a quadratic factor of the cubicovariant ; and considering the three pairs of
sibiconjugates, or what is the same thing, the six linear £Eu;tors of the cubicovariant,
the factors of a pair are the sibiconjugates of the involution formed by the other two
pairs of £Eu;tors.
In fact, the sibiconjugates of the involution formed by the equations
(xay) (xBy) =0, (x fiy)(x  ryy) =
are found by means of the Jacobian of these two functions, viz. of the quadrics
(2, S~a, 2Sa$a?, y)\
(2, )37, 2/37$a:, yY,
which is
(S + ai87, Sa + /3y, Sa(fi + y)^ I3y(8 + a)^x, yY,
viz. a quadratic factor of the cubicovariant; and forming the other two factors, there is
no diflBculty in seeing that any one of these is the Jacobian of the other two.
145. In the case of a pair of equal roots, we have
ai [7= (x ayY {x  yy) (x  By),
a^I ^ tV («7)'(««)',
D = 0,
a»4)= A(7S)»(2a7S, yh^a^ ya^ + Za^2yaZ\x, yYixayy.
70—2
556 A FIFTH MEMOIR UPON QU ANTICS. [156
In the case of two pairs of equal roots, we have
a^ U= {x^ ayy (x  7y)»,
D = 0,
a^J?=^(a7)«(icay)«(a?7y)»,
4>= 0;
these values give also
6IH9JU=0.
146. In the case of three equal roots, we have
a~* U= (x — ayy (x  Sy),
7=0, /=o, n=o,
a'H=^i^(x'Sy{2(x^Byy + {x^ayy}(x^ayy,
and in the case of four equal roots, we have
a"* [7= (a?— cry)*,
7=0, /=o, n=o,
£r=0, 4) = 0.
The preceding formulee, for the case of equal roots, agree with the results obtained
in my memoir on the conditions for the existence of given systems of equalities
between the roots of an equation.
Addition, 7th October, 1858.
Covariant and other Tables (binary quadrics Nos. 25 bis, 29 A, 49 A, and 50 bis).
Mr Salmon has pointed out to me, that in the Table No. 25 of the simplest
octinvariant of a binary quintic^ the coefficients — 210, — 17, + 18 and +38 are
erroneous, and has communicated to me the corrected values, which I have since
verified : the terms, with the corrected values of the coefficients, are [shown in the Table]
No. 25 bis.
[The terms with the erroneous coefficients were alx^cPef, ac^f^, b^d^/*, bc^d^e ; the
correct values —220, —27, +22, and +74 of the coefficients are given in the Table
Q, No. 25, p. 288.]
^ Second Memoir, Philosophical Transactions y t. czlvi. (1S56) p. 125.
156] A FIFTH MEMOIR UPON QU ANTICS, 557
Mr Salmon has also performed the laborious calculation of Hermites' 18thie
invariant of a binary quintic, and has kindly permitted me to publish the result, which
is given in the following Table:
No. 29 a
[This is the Table W No. 29 a given pp. 299—303, the form being slightly
altered as appears p. 282.]
Mr Salmon has also remarked to me, that in the Table No. 50 of the cubin
variant of a binary dodecadicS the coefficients are altogether erroneoua There was, in
fact, a fundamental error in the original calculation; instead of repeating it, I have,
with a view to the deduction therefrom of the cubinvariant (see Fourth Memoir,
No. 78), first calculated the dodecadic quadricovariant, the value of which is given in
the following Table:
No. 49 a
[For this Table see p. 319.]
It is now very easy to obtain the cubinvariant, which is
No. 50 bis.
[This is the Table No. 50, p. 319, the original Na 50 with coefficients which
were altogether erroneous having been omitted.]
1 Third Memoir, Philosophical Trantactiont, t, oxlvi. (1S56) p. 635.
558
[157
157.
ON THE TANGENTIAL OF A CUBIC.
[From the Philosophical Transactions of the Royal Society of London, voL xlviil for the
year 1858, pp. 461—463. Received February 11,— Read March 18, 1858.]
In my "Memoir on Curves of the Third Order *'(*), I had occasion to consider a
derivative which may be termed the "tangential" of a cubic, viz. the tangent at
the point (w, y, z) of the cubic curve (•$ic, y, ^)' = meets the curve in a point
(f , 17, f), which is the tangential of the firstmentioned point ; and I showed that when
the cubic is represented in the canonical form a:^ + y* + z*+ 6lxyz = 0, the coordinates of
the tangential may be taken to be x(y^ — z^) : y (z^ — a^) : z{a?^y^). The method given for
obtaining the tangential may be applied to the general form (a, 6, c,/, g^ h, i^j, k, V$jc, y, zy:
it seems desirable, in reference to the theory of cubic forms, to give the expression of
the tangential for the general form^; and this is what I propose to do, merely indicating
the steps of the calculation, which was performed for me by Mr Greedy.
The cubic foim is
(a, 6, c, /, g, A, i, j, k, l^x, y, z)\
which means
aa^ + by^ + cz^+Sfy^z+Sgz^xh^kr^ + Siysi^ + Sjza^ + Shxf + 6lxyz ;
and the expression for f is obtained from the equation
^f = (^ / *» c$( j, /, c, i, g, l^x, y, z)\  (A, 6, i, /, Z, k^x, y, zy)*
 (a, 6, c, /, flf, A, i, j, k, l^x, y, zy ((Ex + B),
1 Philosophical TramactioM, vol. cxlvii. (1857), [146].
' At the time when the present paper was written, I was not aware of Bir Salmon's theorem (Higher
Plane Curves, p. 156), that the tangential of a point of the cubic is the intersection of the tangent of the
cubic with the first or line polar of the point with respect to the Hessian; a theorem, which at the same
time that it affords the easiest mode of calculation, renders the actual calculation of the coordinates of the
tangential less important. Added 7th October, 1858. — A. G.
157]
ON THE TANGENTIAL OF A CUBIC.
559
where the second line is in fact equal to zero, on account of the first factor, which
vanishes. And GD, Id denote respectively quadric and cubic functions of (y, z\ which
are to be determined so as to make the righthand side divisible by a^\ the resulting
value of f may be modified by the adjunction of the evanescent term
(aj7 H hy ^^z) (a, 6, c, /, g, A, i, j, A?, l\x, y, zf,
where a, h, j are arbitrary coefficients ; but as it is not obvious how these coefficients
should be determined in order to present the result in the most simple form, I have
given the result in the form in which it was obtained without the adjunction of any
such term.
Write for shortness,
R = (i> ff> $y> a^)*
'S=(/, i c $y, zy,
G=(k, I, g Jy, zy,
^ = (t» /. *» c$y, zy,
(A, 6, i, /, I, k $fl?, y, zy^{K P, Q \x, 1)»,
(i, /, c, t, 5r, Z $.;, y, zy = {j, 12, S $^. 1)«,
(a, 6, c, /, flf. A, i, j, Ar, ZJa;, y, zy = (a, B, C, D^x, ly.
GDiT + B =(GD, B $a?, 1),
and then for greater convenience writing (A, 2P, Q\x, ly, &c. for (A, P, Q^x, 1)', &c.,
and omitting the {x, ly, &c. and the arrowheads, or representing the functions simply
by (A, 2P, Q), &c., we have
^f = 6 ( j, 212, £f y
 3/( j, 212, S )» . (A , 2P, Q)
+ 3i(j, 212, £f ).(A, 2P, Qy
 c . (A , 2P, Qy
 (a, 35, 30, D) . (GD, B ),
80 that
which can be developed in terms of the quantities which enter into it. The con
ditions, in order that the coefficients of x, of* may vanish, are thus seen to be '
DB = bS*  3fS'Q + SiSQ'  c(?,
D®  30B = 6 (612S»)  3/(2£f»P + 412SQ) + Si (212(? + 4SPQ)  c (6PQ»),
and from these we obtain
ffi =
6cA:3
/ 6iZ +6
^ fik +3
fH 6
6i^ +6
c/>fc6
r»A: +6
6c^ +3
c/Z6
t^Z +6
\y. «)'
560
ON THE TANGENTIAL OF A CUBIC.
[157
J'c 1
( 6/t +3
/* 2
ftc/3
M» +6
/H3
bei +3
c/'6
fP +3
be' +1
c/t 3
t^ +2
l^y. »)*
and substituting these values, the righthand side of the equation divides by a^, and
throwing out this factor we have the value of f ; and the values of 17, f may be
thence deduced by a mere interchange of letters. The value for f is
X*
^
x«z
xV'
ar^«
x^z*
«/*
xy^t
bp +1
hj'l + 6
6^"* + 6
abck + 3
o^^ — 6
ahcg — 3
a6*c + 1
abcf + 3 ,
c/*'  1
ch^k  6
chH  6
oWZ — 6
acfk + 6
acfl + 6
06/13
a6i«  6
/A/ 3
fhjl 12
/^V 12
afH + 6
a/V+ 6
q/gri + 3
a/» + 2
afH + 3
hHj + 3
^•^^  6
jri  6
o/TA;  3
at^A;  6
ai'l  6
6cAA;  3
6cA/  12
hHl + 6
^/*^i + 6
hch^  3
ft^V +24
6cf + 3
hhU  6
hi^k + 9
hijk +12
hijl +12
hhij + 6
Hf + 6
W +12
hijk +12
6^At — 6
6;Z« +12
cA'  6
c/*A;  6
6^+8
hgl^ +24
chh?  12
chkl  24
M 24
cA»  8
«;7 +18
/%•  6
/»/  6
/i^A 12
/W + 6
c/hk 6
fhH  3
/i^A/  24
/a;'^ 24
Pjk 12
c)fc«Z 24
/feZ« 12
fgjk  24
/V*  3
yAtA; + 3
rgh+ 6
fjkl 24
^^^ 24
^/it7 +24
/W» 24
/!;7 18
AiVfc/ +24
ij]^ +12
ghik + 24
;a^ + 6
hiP +24
yH 4 24
hi^j + 6
t;7« +12
t*»; +24
/^A/ 48
fhU +12
yt;^  9
//» 24
^ +24
A«A: + 6
tA:^ +48
xyz*
xz^
y*
y3^
yV
y^
«^
abet — 3
abc"  1
6«y + 3
6 V + 3
ftc/!; + 9
hc^h  3
hcg^ + 3
ac/^ + 6
ac/i + 3
hcl^  3
hcfh  3
hchi  9
6c^Z + 6
cVA  3
a/i^  3
ai»  2
b/y  3
hckl  3
bgU +18
hcij + 3
c/i^/  6
6c^A  9
6c^' + 3
6/At 3
6/v + 3
c/fcZ 18
6^+6
c/v + 3
bcjl +12
^ + 8
bikl + 6
hgik + 3
/y  18
cPJ + 6
cAt" + 3
6^; +24
cM  12
Ph + 3
6Ai«  6
pa  9
c/^A;  6
/i^  3
bgij + 6
c/!;7 + 3
/»A:;  6
hiP +12
fhi" + 9
cfhi  3
gi^l + 6
c/A^ 18
cP ^ S
/iA:« + 3
c/Ar»  6
t»ifcZ +18
c/^ 12
»!;  3